{"id":2921,"date":"2018-07-26T19:08:48","date_gmt":"2018-07-26T19:08:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/?post_type=chapter&#038;p=2921"},"modified":"2018-07-26T19:08:48","modified_gmt":"2018-07-26T19:08:48","slug":"graphs-of-the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/graphs-of-the-other-trigonometric-functions\/","title":{"raw":"Graphs of the Other Trigonometric Functions","rendered":"Graphs of the Other Trigonometric Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Analyze the graph of \u2009y=tan\u2009x.<\/li>\r\n \t<li>Graph variations of \u2009y=tan\u2009x.<\/li>\r\n \t<li>Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\r\n \t<li>Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\r\n \t<li>Analyze the graph of \u2009y=cot\u2009x.<\/li>\r\n \t<li>Graph variations of \u2009y=cot\u2009x.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1420477\">We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.<\/p>\r\n\r\n<div id=\"fs-id1321556\" class=\"bc-section section\">\r\n<h3>Analyzing the Graph of <em>y<\/em> = tan <em>x<\/em><\/h3>\r\n<p id=\"fs-id1530665\">We will begin with the graph of the <span class=\"no-emphasis\">tangent<\/span> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\r\n\r\n<div id=\"fs-id1614888\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\,x=\\frac{\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}[\/latex]<\/div>\r\n<p id=\"fs-id1288893\">The <span class=\"no-emphasis\">period<\/span> of the tangent function is[latex]\\,\\pi \\,[\/latex]because the graph repeats itself on intervals of[latex]\\,k\\pi \\,[\/latex]where[latex]\\,k\\,[\/latex]is a constant. If we graph the tangent function on[latex]\\,-\\frac{\\pi }{2}\\,[\/latex]to[latex]\\,\\frac{\\pi }{2},\\,[\/latex]we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.<\/p>\r\n<p id=\"fs-id1647294\">We can determine whether tangent is an odd or even function by using the definition of tangent.<\/p>\r\n\r\n<div id=\"fs-id1615402\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\mathrm{tan}\\left(-x\\right)=\\frac{\\mathrm{sin}\\left(-x\\right)}{\\mathrm{cos}\\left(-x\\right)}\\hfill &amp; \\begin{array}{ccc}&amp; &amp; \\end{array}\\text{Definition of tangent}.\\hfill \\\\ \\text{ }=\\frac{-\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}\\hfill &amp; \\begin{array}{ccc}&amp; &amp; \\end{array}\\text{Sine is an odd function, cosine is even}.\\hfill \\\\ \\text{ }=-\\frac{\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}\\hfill &amp; \\begin{array}{ccc}&amp; &amp; \\end{array}\\text{The quotient of an odd and an even function is odd}.\\hfill \\\\ \\text{ }=-\\mathrm{tan}\\,x\\hfill &amp; \\begin{array}{ccc}&amp; &amp; \\end{array}\\text{Definition of tangent}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1460636\">Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in <a class=\"autogenerated-content\" href=\"#Table_06_02_00\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_06_02_00\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]-\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td>[latex]-\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]-\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td>[latex]-\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td>undefined<\/td>\r\n<td>[latex]-\\sqrt{3}[\/latex]<\/td>\r\n<td>\u20131<\/td>\r\n<td>[latex]-\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1279869\">These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when[latex]\\,\\frac{\\pi }{3}&lt;x&lt;\\frac{\\pi }{2},\\,[\/latex]we can use a table to look for a trend. Because[latex]\\,\\frac{\\pi }{3}\\approx 1.05\\,[\/latex]and[latex]\\,\\frac{\\pi }{2}\\approx 1.57,\\,[\/latex]we will evaluate[latex]\\,x\\,[\/latex]at radian measures[latex]\\,1.05&lt;x&lt;1.57\\,[\/latex]as shown in <a class=\"autogenerated-content\" href=\"#Table_06_02_01\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>1.3<\/td>\r\n<td>1.5<\/td>\r\n<td>1.55<\/td>\r\n<td>1.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]\\mathrm{tan} \\text{ }x[\/latex]<\/strong><\/td>\r\n<td>3.6<\/td>\r\n<td>14.1<\/td>\r\n<td>48.1<\/td>\r\n<td>92.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1658883\">As[latex]\\,x\\,[\/latex]approaches[latex]\\,\\frac{\\pi }{2},\\,[\/latex]the outputs of the function get larger and larger. Because[latex]\\,y=\\mathrm{tan}\\,x\\,[\/latex]is an odd function, we see the corresponding table of negative values in <a class=\"autogenerated-content\" href=\"#Table_06_02_02\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22121.3<\/td>\r\n<td>\u22121.5<\/td>\r\n<td>\u22121.55<\/td>\r\n<td>\u22121.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]\\mathrm{tan}\\,x[\/latex]<\/strong><\/td>\r\n<td>\u22123.6<\/td>\r\n<td>\u221214.1<\/td>\r\n<td>\u221248.1<\/td>\r\n<td>\u221292.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1318044\">We can see that, as[latex]\\,x\\,[\/latex]approaches[latex]\\,-\\frac{\\pi }{2},\\,[\/latex]the outputs get smaller and smaller. Remember that there are some values of[latex]\\,x\\,[\/latex]for which[latex]\\,\\mathrm{cos}\\,x=0.\\,[\/latex]For example,[latex]\\,\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0\\,[\/latex]and[latex]\\,\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0.\\,[\/latex]At these values, the <span class=\"no-emphasis\">tangent function<\/span> is undefined, so the graph of[latex]\\,y=\\mathrm{tan}\\,x\\,[\/latex]has discontinuities at[latex]\\,x=\\frac{\\pi }{2}\\text{ and }\\frac{3\\pi }{2}.\\,[\/latex]At these values, the graph of the tangent has vertical asymptotes. <a class=\"autogenerated-content\" href=\"#Figure_06_02_001\">(Figure)<\/a> represents the graph of[latex]\\,y=\\mathrm{tan}\\,x.\\,[\/latex]The tangent is positive from 0 to[latex]\\,\\frac{\\pi }{2}\\,[\/latex]and from[latex]\\,\\pi \\,[\/latex]to[latex]\\,\\frac{3\\pi }{2},\\,[\/latex]corresponding to quadrants I and III of the unit circle.<\/p>\r\n\r\n<div id=\"Figure_06_02_001\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143550\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/> <strong>Figure 1. <\/strong>Graph of the tangent function[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1678025\" class=\"bc-section section\">\r\n<h3>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h3>\r\n<p id=\"fs-id1232225\">As with the sine and cosine functions, the <span class=\"no-emphasis\">tangent<\/span> function can be described by a general equation.<\/p>\r\n\r\n<div id=\"fs-id988668\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{tan}\\left(Bx\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1393080\">We can identify horizontal and vertical stretches and compressions using values of[latex]\\,A\\,[\/latex]and[latex]\\,B.\\,[\/latex]The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.<\/p>\r\n<p id=\"fs-id933588\">Because there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant[latex]\\,A.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1409849\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\r\n<ul id=\"fs-id1300606\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is all real numbers[latex]\\,x,[\/latex]where[latex]\\,x\\ne \\frac{\\pi }{2|B|}+\\frac{\\pi }{|B|}k\\,[\/latex]such that[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\r\n \t<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{2|B|}+\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>[latex]y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex]is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1430247\" class=\"bc-section section\">\r\n<h4>Graphing One Period of a Stretched or Compressed Tangent Function<\/h4>\r\n<p id=\"fs-id1698843\">We can use what we know about the properties of the <span class=\"no-emphasis\">tangent function<\/span> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(Bx\\right).\\,[\/latex]We focus on a single <span class=\"no-emphasis\">period<\/span> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval[latex]\\,\\left(-\\frac{P}{2},\\frac{P}{2}\\right)\\,[\/latex]and the graph has vertical asymptotes at[latex]\\,\u00b1\\frac{P}{2}\\,[\/latex]where[latex]\\,P=\\frac{\\pi }{B}.\\,[\/latex]On[latex]\\,\\left(-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right),\\,[\/latex]the graph will come up from the left asymptote at[latex]\\,x=-\\frac{\\pi }{2},\\,[\/latex]cross through the origin, and continue to increase as it approaches the right asymptote at[latex]\\,x=\\frac{\\pi }{2}.\\,[\/latex]To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use<\/p>\r\n\r\n<div id=\"fs-id1631239\" class=\"unnumbered aligncenter\">[latex]f\\left(\\frac{P}{4}\\right)=A\\mathrm{tan}\\left(B\\frac{P}{4}\\right)=A\\mathrm{tan}\\left(B\\frac{\\pi }{4B}\\right)=A[\/latex]<\/div>\r\n<p id=\"fs-id1638091\">because[latex]\\,\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1644376\" class=\"precalculus howto\">\r\n<p id=\"fs-id917405\"><strong>Given the function[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1270463\" type=\"1\">\r\n \t<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Draw vertical asymptotes at[latex]\\,x=-\\frac{P}{2}\\,[\/latex]and[latex]\\,x=\\frac{P}{2}.[\/latex]<\/li>\r\n \t<li>For[latex]\\,A&gt;0,\\,[\/latex]the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for[latex]\\,A&lt;0[\/latex]).<\/li>\r\n \t<li>Plot reference points at[latex]\\,\\left(\\frac{P}{4},A\\right),\\,[\/latex][latex]\\left(0,0\\right),\\,[\/latex]and[latex]\\,\\left(-\\frac{P}{4},-A\\right),\\,[\/latex]and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1426111\">\r\n<div id=\"fs-id1678052\">\r\n<h3>Sketching a Compressed Tangent<\/h3>\r\n<p id=\"fs-id1316312\">Sketch a graph of one period of the function[latex]\\,y=0.5\\mathrm{tan}\\left(\\frac{\\pi }{2}x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1051354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1051354\"]\r\n<p id=\"fs-id1051354\">First, we identify[latex]\\,A\\,[\/latex]and[latex]\\,B.[\/latex]<\/p>\r\n<span id=\"fs-id1654424\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143600\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" \/><\/span>\r\n<p id=\"fs-id1701807\">Because[latex]\\,A=0.5\\,[\/latex]and[latex]\\,B=\\frac{\\pi }{2},\\,[\/latex]we can find the <span class=\"no-emphasis\">stretching\/compressing factor<\/span> and period. The period is[latex]\\,\\frac{\\pi }{\\frac{\\pi }{2}}=2,\\,[\/latex]so the asymptotes are at[latex]\\,x=\u00b11.\\,[\/latex]At a quarter period from the origin, we have<\/p>\r\n\r\n<div id=\"fs-id1061460\">[latex]\\begin{array}{l}f\\left(0.5\\right)=0.5\\mathrm{tan}\\left(\\frac{0.5\\pi }{2}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.5\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.5\\hfill \\end{array}[\/latex]<\/div>\r\nThis means the curve must pass through the points[latex]\\,\\left(0.5,0.5\\right),[\/latex][latex]\\left(0,0\\right),[\/latex]and[latex]\\,\\left(-0.5,-0.5\\right).\\,[\/latex]The only inflection point is at the origin. <a class=\"autogenerated-content\" href=\"#Figure_06_02_003\">(Figure)<\/a> shows the graph of one period of the function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143606\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/> <strong>Figure 2.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1009376\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_02\">\r\n<div id=\"fs-id1706988\">\r\n<p id=\"fs-id1706989\">Sketch a graph of[latex]\\,f\\left(x\\right)=3\\mathrm{tan}\\left(\\frac{\\pi }{6}x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1354903\"]Show Solution[\/reveal-answer][hidden-answer a=\"1354903\"]<span id=\"fs-id1439999\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143619\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1464435\" class=\"bc-section section\">\r\n<h4>Graphing One Period of a Shifted Tangent Function<\/h4>\r\n<p id=\"fs-id1406087\">Now that we can graph a <span class=\"no-emphasis\">tangent function<\/span> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add[latex]\\,C\\,[\/latex]and[latex]\\,D\\,[\/latex]to the general form of the tangent function.<\/p>\r\n\r\n<div id=\"fs-id1270990\">[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(Bx-C\\right)+D[\/latex]<\/div>\r\n<p id=\"fs-id1241838\">The graph of a transformed tangent function is different from the basic tangent function[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]in several ways:<\/p>\r\n\r\n<div id=\"eip-87\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\r\n<ul id=\"fs-id1346106\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\r\n \t<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\,\\mathrm{tan}\\left(Bx-C\\right)+D\\,[\/latex]is an odd function because it is the quotient of odd and even functions (sin and cosine respectively).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1665485\" class=\"precalculus howto\">\r\n<p id=\"fs-id1432077\"><strong>Given the function[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D,\\,[\/latex]sketch the graph of one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1459436\" type=\"1\">\r\n \t<li>Express the function given in the form[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li>Identify the <span class=\"no-emphasis\">stretching\/compressing factor<\/span>,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex]shifted to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at[latex]\\text{ }x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\text{ }k\\text{ }[\/latex]is an odd integer.<\/li>\r\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1677226\">\r\n<div id=\"fs-id1663685\">\r\n<h3>Graphing One Period of a Shifted Tangent Function<\/h3>\r\n<p id=\"fs-id1355894\">Graph one period of the function[latex]\\,y=-2\\mathrm{tan}\\left(\\pi x+\\pi \\right)\\,-1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"826718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"826718\"]\r\n<ul id=\"eip-id1165133446963\">\r\n \t<li><em>Step 1.<\/em> The function is already written in the form[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em>[latex]\\,A=-2,\\,[\/latex]so the stretching factor is[latex]\\,|A|=2.[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em>[latex]\\,B=\\pi ,\\,[\/latex]so the period is[latex]\\,P=\\frac{\\pi }{|B|}=\\frac{\\pi }{\\pi }=1.[\/latex]<\/li>\r\n \t<li><em>Step 4.<\/em>[latex]\\,C=-\\pi ,\\,[\/latex]so the phase shift is[latex]\\,\\frac{C}{B}=\\frac{-\\pi }{\\pi }=-1.[\/latex]<\/li>\r\n \t<li><em>Step 5-7.<\/em> The asymptotes are at[latex]\\,x=-\\frac{3}{2}\\,[\/latex]and[latex]\\,x=-\\frac{1}{2}\\,[\/latex]and the three recommended reference points are[latex]\\,\\left(-1.25,1\\right),\\,[\/latex][latex]\\left(-1,-1\\right),\\,[\/latex]and[latex]\\,\\left(-0.75,-3\\right).\\,[\/latex]The graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_005\">(Figure)<\/a>.\r\n<div id=\"Figure_06_02_005\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143622\/CNX_Precalc_Figure_06_02_005.jpg\" alt=\"A graph of one period of a shifted tangent function, with vertical asymptotes at x=-1.5 and x=-0.5.\" width=\"487\" height=\"193\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1700370\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1456051\">Note that this is a decreasing function because[latex]\\,A&lt;0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1586707\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_03\">\r\n<div id=\"fs-id1379854\">\r\n<p id=\"fs-id1379856\">How would the graph in <a class=\"autogenerated-content\" href=\"#Example_06_02_02\">(Figure)<\/a> look different if we made[latex]\\,A=2\\,[\/latex]instead of[latex]\\,-2?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1394906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1394906\"]\r\n<p id=\"fs-id1394906\">It would be reflected across the line[latex]\\,y=-1,\\,[\/latex]becoming an increasing function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id672550\" class=\"precalculus howto\">\r\n<p id=\"fs-id1422588\"><strong>Given the graph of a tangent function, identify horizontal and vertical stretches.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1586539\" type=\"1\">\r\n \t<li>Find the period[latex]\\,P\\,[\/latex]from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\r\n \t<li>Write[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{P}x\\right).[\/latex]<\/li>\r\n \t<li>Determine a convenient point[latex]\\,\\left(x,f\\left(x\\right)\\right)\\,[\/latex]on the given graph and use it to determine[latex]\\,A.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_03\" class=\"textbox examples\">\r\n<div>\r\n<div id=\"fs-id1430638\">\r\n<h3>Identifying the Graph of a Stretched Tangent<\/h3>\r\n<p id=\"fs-id1586016\">Find a formula for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_06_02_006\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_02_006\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143625\/CNX_Precalc_Figure_06_02_006.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.\" width=\"487\" height=\"256\" \/> <strong>Figure 4. <\/strong>A stretched tangent function[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1649014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1649014\"]\r\n<p id=\"fs-id1649014\">The graph has the shape of a tangent function.<\/p>\r\n\r\n<ul id=\"eip-id1165134188910\">\r\n \t<li><em>Step 1.<\/em> One cycle extends from \u20134 to 4, so the period is[latex]\\,P=8.\\,[\/latex]Since[latex]\\,P=\\frac{\\pi }{|B|},\\,[\/latex]we have[latex]\\,B=\\frac{\\pi }{P}=\\frac{\\pi }{8}.[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em> The equation must have the form[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{8}x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em> To find the vertical stretch[latex]\\,A,[\/latex]we can use the point[latex]\\,\\left(2,2\\right).[\/latex]\r\n<div id=\"fs-id638950\" class=\"unnumbered aligncenter\">[latex]2=A\\mathrm{tan}\\left(\\frac{\\pi }{8}\\cdot 2\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/div><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1536654\">Because[latex]\\,\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1,\\,[\/latex][latex]A=2.[\/latex]<\/p>\r\n<p id=\"fs-id1409295\">This function would have a formula[latex]\\,f\\left(x\\right)=2\\mathrm{tan}\\left(\\frac{\\pi }{8}x\\right).[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1655580\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_04\">\r\n<div id=\"fs-id1563795\">\r\n<p id=\"fs-id1563796\">Find a formula for the function in <a class=\"autogenerated-content\" href=\"#Figure_06_02_007\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_02_007\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143628\/CNX_Precalc_Figure_06_02_007.jpg\" alt=\"A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi\/4, -pi\/4, pi\/4, and 3pi\/4.\" width=\"487\" height=\"315\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1662824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1662824\"]\r\n<p id=\"fs-id1662824\">[latex]g\\left(x\\right)=4\\mathrm{tan}\\left(2x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1653007\" class=\"bc-section section\">\r\n<h3>Analyzing the Graphs of <em>y<\/em> = sec <em>x<\/em> and <em>y<\/em> = csc<em>x<\/em><\/h3>\r\n<p id=\"fs-id1405395\">The <span class=\"no-emphasis\">secant<\/span> was defined by the <span class=\"no-emphasis\">reciprocal identity<\/span>[latex]\\,\\mathrm{sec}\\,x=\\frac{1}{\\mathrm{cos}\\,x}.\\,[\/latex]Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at[latex]\\,\\frac{\\pi }{2},\\,[\/latex][latex]\\frac{3\\pi }{2},\\,[\/latex]etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\r\n<p id=\"fs-id1615608\">We can graph[latex]\\,y=\\mathrm{sec}\\,x\\,[\/latex]by observing the graph of the cosine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_008\">(Figure)<\/a>. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <span class=\"no-emphasis\">secant function<\/span> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.<\/p>\r\n<p id=\"fs-id1657904\">The secant graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.<\/p>\r\n<p id=\"fs-id1457737\">Note that, because cosine is an even function, secant is also an even function. That is,[latex]\\,\\mathrm{sec}\\left(-x\\right)=\\mathrm{sec}\\,x.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_02_008\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143637\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" width=\"487\" height=\"379\" \/> <strong>Figure 6. <\/strong>Graph of the secant function,[latex]\\,f\\left(x\\right)=\\mathrm{sec}x=\\frac{1}{\\mathrm{cos}x}[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1584466\">As we did for the tangent function, we will again refer to the constant[latex]\\,|A|\\,[\/latex]as the stretching factor, not the amplitude.<\/p>\r\n\r\n<div id=\"fs-id1700128\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\r\n<ul id=\"fs-id1440349\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{2|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(-\\infty ,-|A|\\right]\\cup \\left[|A|,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The vertical asymptotes occur at[latex]\\,x=\\frac{\\pi }{2|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\mathrm{sec}\\left(Bx\\right)\\,[\/latex]is an even function because cosine is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1380717\">Similar to the secant, the <span class=\"no-emphasis\">cosecant<\/span> is defined by the reciprocal identity[latex]\\,\\mathrm{csc}\\,x=\\frac{1}{\\mathrm{sin}\\,x}.\\,[\/latex]Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at[latex]\\,0,\\,[\/latex][latex]\\pi ,\\,[\/latex]etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\r\n<p id=\"fs-id1413697\">We can graph[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]by observing the graph of the sine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">(Figure)<\/a>. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <span class=\"no-emphasis\">cosecant function<\/span> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\r\n<p id=\"fs-id1380921\">The cosecant graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\r\n<p id=\"fs-id1457402\">Note that, since sine is an odd function, the cosecant function is also an odd function. That is,[latex]\\,\\mathrm{csc}\\left(-x\\right)=\\mathrm{-csc}x.[\/latex]<\/p>\r\n<p id=\"eip-610\">The graph of cosecant, which is shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">(Figure)<\/a>, is similar to the graph of secant.<\/p>\r\n\r\n<div id=\"Figure_06_02_009\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143642\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" width=\"487\" height=\"377\" \/> <strong>Figure 7. <\/strong>The graph of the cosecant function,[latex]\\,f\\left(x\\right)=\\mathrm{csc}x=\\frac{1}{\\mathrm{sin}x}[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1683902\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>csc(<em>Bx<\/em>)<\/h3>\r\n<ul id=\"fs-id1425299\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\left(-\\infty ,-|A|\\right]\\cup \\left[|A|,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>[latex]y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]is an odd function because sine is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1670966\" class=\"bc-section section\">\r\n<h3>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y<\/em>= csc <em>x<\/em><\/h3>\r\n<p id=\"fs-id1420053\">For shifted, compressed, and\/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the <span class=\"no-emphasis\">cosecant function<\/span> in the same way as for the secant and other functions.The equations become the following.<\/p>\r\n\r\n<div id=\"eip-341\">[latex]y=A\\mathrm{sec}\\left(Bx-C\\right)+D[\/latex]<\/div>\r\n<div>[latex]y=A\\mathrm{csc}\\left(Bx-C\\right)+D[\/latex]<\/div>\r\n<div id=\"eip-992\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\r\n<ul id=\"eip-id2701214\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]is an even function because cosine is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"eip-439\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>csc(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\r\n<ul id=\"eip-id1449110\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,[\/latex]is an odd function because sine is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1420056\" class=\"precalculus howto\">\r\n<p id=\"fs-id1512502\"><strong>Given a function of the form[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1690218\" type=\"1\">\r\n \t<li>Express the function given in the form[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Sketch the graph of[latex]\\,y=A\\mathrm{cos}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Use the reciprocal relationship between[latex]\\,y=\\mathrm{cos}\\,x\\,[\/latex]and[latex]\\,y=\\mathrm{sec}\\,x\\,[\/latex]to draw the graph of[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1698723\">\r\n<div id=\"fs-id1406755\">\r\n<h3>Graphing a Variation of the Secant Function<\/h3>\r\n<p id=\"fs-id1325018\">Graph one period of[latex]\\,f\\left(x\\right)=2.5\\mathrm{sec}\\left(0.4x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"331336\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"331336\"]\r\n<ul id=\"eip-id1165135426343\">\r\n \t<li><em>Step 1.<\/em> The given function is already written in the general form,[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em>[latex]\\,A=2.5\\,[\/latex]so the stretching factor is[latex]\\,\\text{2}\\text{.5}\\text{.}[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em>[latex]\\,B=0.4\\,[\/latex]so[latex]\\,P=\\frac{2\\pi }{0.4}=5\\pi .\\,[\/latex]The period is[latex]\\,5\\pi \\,[\/latex]units.<\/li>\r\n \t<li><em>Step 4.<\/em> Sketch the graph of the function[latex]\\,g\\left(x\\right)=2.5\\mathrm{cos}\\left(0.4x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 5.<\/em> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.<\/li>\r\n \t<li><em>Steps 6\u20137.<\/em> Sketch two asymptotes at[latex]\\,x=1.25\\pi \\,[\/latex]and[latex]\\,x=3.75\\pi .\\,[\/latex]We can use two reference points, the local minimum at[latex]\\,\\left(0,2.5\\right)\\,[\/latex]and the local maximum at[latex]\\,\\left(2.5\\pi ,-2.5\\right).\\,[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_010\">(Figure)<\/a> shows the graph.\r\n<div id=\"Figure_06_02_010\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143645\/CNX_Precalc_Figure_06_02_010.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an upward facing prarbola and a downward facing parabola.\" width=\"487\" height=\"567\" \/> <strong>Figure 8.<\/strong>[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1439628\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_05\">\r\n<div id=\"fs-id1690764\">\r\n<p id=\"fs-id1354617\">Graph one period of[latex]\\,f\\left(x\\right)=-2.5\\mathrm{sec}\\left(0.4x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1440114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1440114\"]\r\n<p id=\"fs-id1440114\">This is a vertical reflection of the preceding graph because[latex]\\,A\\,[\/latex]is negative.<\/p>\r\n<span id=\"fs-id1380832\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143649\/CNX_Precalc_Figure_06_02_011.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1570004\" class=\"precalculus qa textbox shaded\">\r\n<p id=\"fs-id1440136\"><strong>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/strong><\/p>\r\n<p id=\"fs-id1420221\"><em>Yes. The range of<\/em>[latex]\\,f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]<em>is<\/em>[latex]\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1638418\" class=\"precalculus howto\">\r\n\r\n<strong>Given a function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong>\r\n<ol id=\"eip-id2515230\" type=\"1\">\r\n \t<li>Express the function given in the form[latex]\\,y=A\\,\\mathrm{sec}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\,\\mathrm{sec}\\left(Bx\\right)\\,.[\/latex]but shift it to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1431671\">\r\n<div id=\"fs-id1431673\">\r\n<h3>Graphing a Variation of the Secant Function<\/h3>\r\n<p id=\"fs-id1537180\">Graph one period of[latex]\\,y=4\\mathrm{sec}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{2}\\right)+1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"318489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"318489\"]\r\n<ul id=\"fs-id1906405\">\r\n \t<li><em>Step 1.<\/em> Express the function given in the form[latex]\\,y=4\\mathrm{sec}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{2}\\right)+1.[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em> The stretching\/compressing factor is[latex]|A|=4.[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em> The period is\r\n<div id=\"fs-id2071183\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{3}}\\hfill \\\\ \\text{ }=\\frac{2\\pi }{1}\\cdot \\frac{3}{\\pi }\\hfill \\\\ \\text{ }=6\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li><em>Step 4.<\/em> The phase shift is\r\n<div id=\"fs-id2341189\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{C}{B}=\\frac{\\frac{\\pi }{2}}{\\frac{\\pi }{3}}\\hfill \\\\ \\text{ }=\\frac{\\pi }{2}\\cdot \\frac{3}{\\pi }\\hfill \\\\ \\text{ }=1.5\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li><em>Step 5.<\/em> Draw the graph of [latex]\\,y=A\\mathrm{sec}\\left(Bx\\right),[\/latex]but shift it to the right by[latex]\\,\\frac{C}{B}=1.5\\,[\/latex]and up by[latex]\\,D=6.[\/latex]<\/li>\r\n \t<li><em>Step 6.<\/em> Sketch the vertical asymptotes, which occur at[latex]\\,x=0,x=3,[\/latex]and[latex]\\,x=6.\\,[\/latex]There is a local minimum at[latex]\\,\\left(1.5,5\\right)\\,[\/latex]and a local maximum at[latex]\\,\\left(4.5,-3\\right).\\,[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_012\">(Figure)<\/a> shows the graph.<\/li>\r\n<\/ul>\r\n<div id=\"Figure_06_02_012\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143653\/CNX_Precalc_Figure_06_02_012.jpg\" alt=\"\" width=\"487\" height=\"318\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1451968\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_06\">\r\n<div id=\"fs-id1346086\">\r\n<p id=\"fs-id1410617\">Graph one period of[latex]\\,f\\left(x\\right)=-6\\mathrm{sec}\\left(4x+2\\right)-8.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1439474\"]Show Solution[\/reveal-answer][hidden-answer a=\"1439474\"]<span id=\"fs-id1673799\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143655\/CNX_Precalc_Figure_06_02_013.jpg\" alt=\"A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi\/20 and one approximately at 3pi\/16.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1523838\" class=\"precalculus qa textbox shaded\">\r\n<p id=\"eip-id1165135487331\"><strong>The domain of[latex]\\,\\mathrm{csc}\\,x\\,[\/latex]was given to be all[latex]\\,x\\,[\/latex]such that[latex]\\,x\\ne k\\pi \\,[\/latex]for any integer[latex]\\,k.\\,[\/latex]<strong>Would the domain of<\/strong>[latex]\\,y=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,\\text{be}\\,x\\ne \\frac{C+k\\pi }{B}?[\/latex]<\/strong><\/p>\r\n<p id=\"fs-id1374943\"><em>Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function\u2019s input.<\/em><\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1428931\" class=\"precalculus howto\">\r\n<p id=\"fs-id1629286\"><strong>Given a function of the form[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1395174\" type=\"1\">\r\n \t<li>Express the function given in the form[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Use the reciprocal relationship between[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]and[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]to draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1410488\">\r\n<div id=\"fs-id1410490\">\r\n<h3>Graphing a Variation of the Cosecant Function<\/h3>\r\n<p id=\"fs-id1425806\">Graph one period of[latex]\\,f\\left(x\\right)=-3\\mathrm{csc}\\left(4x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"990021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"990021\"]\r\n<ul id=\"eip-id1165131907381\">\r\n \t<li><em>Step 1.<\/em> The given function is already written in the general form,[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em>[latex]\\,|A|=|-3|=3,[\/latex]so the stretching factor is 3.<\/li>\r\n \t<li><em>Step 3.<\/em>[latex]\\,B=4,[\/latex]so[latex]\\,P=\\frac{2\\pi }{4}=\\frac{\\pi }{2}.\\,[\/latex]The period is[latex]\\,\\frac{\\pi }{2}\\,[\/latex]units.<\/li>\r\n \t<li><em>Step 4.<\/em> Sketch the graph of the function[latex]\\,g\\left(x\\right)=-3\\mathrm{sin}\\left(4x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 5.<\/em> Use the reciprocal relationship of the sine and cosecant functions to draw the <span class=\"no-emphasis\">cosecant function<\/span>.<\/li>\r\n \t<li><em>Steps 6\u20137.<\/em> Sketch three asymptotes at[latex]\\,x=0,\\,x=\\frac{\\pi }{4},\\,[\/latex]and[latex]\\,x=\\frac{\\pi }{2}.\\,[\/latex]We can use two reference points, the local maximum at[latex]\\,\\left(\\frac{\\pi }{8},-3\\right)\\,[\/latex]and the local minimum at[latex]\\,\\left(\\frac{3\\pi }{8},3\\right).[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_023\">(Figure)<\/a> shows the graph.\r\n<div id=\"Figure_06_02_023\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143703\/CNX_Precalc_Figure_06_02_023.jpg\" alt=\"A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi\/4, and x=pi\/2.\" width=\"487\" height=\"686\" \/> <strong>Figure 10.<\/strong>[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1714936\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_07\">\r\n<div id=\"fs-id1422180\">\r\n<p id=\"fs-id1422181\">Graph one period of[latex]\\,f\\left(x\\right)=0.5\\mathrm{csc}\\left(2x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1150397\"]Show Solution[\/reveal-answer][hidden-answer a=\"1150397\"]<span id=\"fs-id1696881\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143715\/CNX_Precalc_Figure_06_02_023b.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id978295\" class=\"precalculus howto\">\r\n<p id=\"fs-id1046161\"><strong>Given a function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{csc}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"eip-id1453973\" type=\"1\">\r\n \t<li>Express the function given in the form[latex]\\,y=A\\mathrm{csc}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by[latex]\\,D.[\/latex]<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1439464\">\r\n<div id=\"fs-id1439466\">\r\n<h3>Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\r\n<p id=\"fs-id1107734\">Sketch a graph of[latex]\\,y=2\\mathrm{csc}\\left(\\frac{\\pi }{2}x\\right)+1.\\,[\/latex]What are the domain and range of this function?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n[reveal-answer q=\"61156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"61156\"]\r\n<ul id=\"eip-id3155416\">\r\n \t<li><em>Step 1.<\/em> Express the function given in the form[latex]\\,y=2\\mathrm{csc}\\left(\\frac{\\pi }{2}x\\right)+1.[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em> Identify the stretching\/compressing factor,[latex]\\,|A|=2.[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em> The period is[latex]\\,\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{2}}=\\frac{2\\pi }{1}\\cdot \\frac{2}{\\pi }=4.[\/latex]<\/li>\r\n \t<li><em>Step 4.<\/em> The phase shift is[latex]\\,\\frac{0}{\\frac{\\pi }{2}}=0.[\/latex]<\/li>\r\n \t<li><em>Step 5.<\/em> Draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]but shift it up[latex]\\,D=1.[\/latex]<\/li>\r\n \t<li><em>Step 6.<\/em> Sketch the vertical asymptotes, which occur at[latex]\\,x=0,x=2,x=4.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id869117\">The graph for this function is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_014\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_02_014\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143731\/CNX_Precalc_Figure_06_02_014F.jpg\" alt=\"A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.\" width=\"487\" height=\"377\" \/> <strong>Figure 11. <\/strong>A transformed cosecant function[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]<\/div>\r\n<div id=\"fs-id1520970\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id728453\">The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of[latex]\\,f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)+1,[\/latex]shown as the orange dashed wave.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1588757\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_06_02_08\">\r\n<div id=\"fs-id1464306\">\r\n<p id=\"fs-id1464307\">Given the graph of[latex]\\,f\\left(x\\right)=2\\mathrm{cos}\\left(\\frac{\\pi }{2}x\\right)+1\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_015\">(Figure)<\/a>, sketch the graph of[latex]\\,g\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{2}x\\right)+1\\,[\/latex]on the same axes.<\/p>\r\n\r\n<div id=\"Figure_06_02_015\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143734\/CNX_Precalc_Figure_06_02_015.jpg\" alt=\"A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.\" width=\"488\" height=\"381\" \/> <strong>Figure 12.<\/strong>[\/caption]<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1528623\"]Show Solution[\/reveal-answer][hidden-answer a=\"1528623\"]<span id=\"fs-id1662489\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143754\/CNX_Precalc_Figure_06_02_016.jpg\" alt=\"A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1232916\" class=\"bc-section section\">\r\n<h3>Analyzing the Graph of <em>y<\/em> = cot <em>x<\/em><\/h3>\r\n<p id=\"fs-id1585113\">The last trigonometric function we need to explore is <span class=\"no-emphasis\">cotangent<\/span>. The cotangent is defined by the <span class=\"no-emphasis\">reciprocal identity<\/span>[latex]\\,\\mathrm{cot}\\,x=\\frac{1}{\\mathrm{tan}\\,x}.\\,[\/latex]Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at[latex]\\,0,\\pi ,\\,[\/latex]etc. Since the output of the tangent function is all real numbers, the output of the <span class=\"no-emphasis\">cotangent function<\/span> is also all real numbers.<\/p>\r\n<p id=\"fs-id1333454\">We can graph[latex]\\,y=\\mathrm{cot}\\,x\\,[\/latex]by observing the graph of the tangent function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_017\">(Figure)<\/a>. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\r\n<p id=\"fs-id1430801\">The cotangent graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x=0;\\,[\/latex]we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,[latex]\\,\\mathrm{cot}\\,x\\,[\/latex]has vertical asymptotes at all values of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x=0,\\,[\/latex]and[latex]\\,\\mathrm{cot}\\,x=0\\,[\/latex]at all values of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]has its vertical asymptotes.<\/p>\r\n\r\n<div id=\"Figure_06_02_017\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143757\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"487\" height=\"439\" \/> <strong>Figure 13. <\/strong>The cotangent function[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1283383\">\r\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\r\n<ul id=\"fs-id1393165\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\r\n \t<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>[latex]y=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1352247\" class=\"bc-section section\">\r\n<h3>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h3>\r\n<p id=\"fs-id1519587\">We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.<\/p>\r\n\r\n<div id=\"eip-1\">[latex]y=A\\mathrm{cot}\\left(Bx-C\\right)+D[\/latex]<\/div>\r\n<div id=\"fs-id1658740\">\r\n<h3>Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\r\n<ul id=\"eip-id3983675\">\r\n \t<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>The period is[latex]\\,\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\r\n \t<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1526877\" class=\"precalculus howto\">\r\n<p id=\"fs-id1674147\"><strong>Given a modified cotangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right),[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id988921\" type=\"1\">\r\n \t<li>Express the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Plot any two reference points.<\/li>\r\n \t<li>Use the reciprocal relationship between tangent and cotangent to draw the graph of[latex]\\,y=A\\mathrm{cot}\\left(Bx\\right).[\/latex]<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_08\" class=\"textbox examples\">\r\n<div id=\"fs-id918576\">\r\n<div id=\"fs-id918578\">\r\n<h3>Graphing Variations of the Cotangent Function<\/h3>\r\n<p id=\"fs-id1422599\">Determine the stretching factor, period, and phase shift of[latex]\\,y=3\\mathrm{cot}\\left(4x\\right),\\,[\/latex]and then sketch a graph.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"526668\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"526668\"]\r\n<ul id=\"eip-id1165137777630\">\r\n \t<li><em>Step 1.<\/em> Expressing the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]gives[latex]\\,f\\left(x\\right)=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em> The stretching factor is[latex]\\,|A|=3.[\/latex]<\/li>\r\n \t<li><em>Step 3.<\/em> The period is[latex]\\,P=\\frac{\\pi }{4}.[\/latex]<\/li>\r\n \t<li><em>Step 4.<\/em> Sketch the graph of[latex]\\,y=3\\mathrm{tan}\\left(4x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 5.<\/em> Plot two reference points. Two such points are[latex]\\,\\left(\\frac{\\pi }{16},3\\right)\\,[\/latex]and[latex]\\,\\left(\\frac{3\\pi }{16},-3\\right).[\/latex]<\/li>\r\n \t<li><em>Step 6.<\/em> Use the reciprocal relationship to draw[latex]\\,y=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/li>\r\n \t<li><em>Step 7.<\/em> Sketch the asymptotes,[latex]\\,x=0,\\,\\,x=\\frac{\\pi }{4}.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1377143\">The orange graph in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_019\">(Figure)<\/a> shows[latex]\\,y=3\\mathrm{tan}\\left(4x\\right)\\,[\/latex]and the blue graph shows[latex]\\,y=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_02_019\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143801\/CNX_Precalc_Figure_06_02_019.jpg\" alt=\"A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi\/4 and pi\/4.\" width=\"487\" height=\"592\" \/> <strong>Figure 14.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1508070\" class=\"precalculus howto\">\r\n<p id=\"fs-id1692984\"><strong>Given a modified cotangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1425287\" type=\"1\">\r\n \t<li>Express the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\r\n \t<li>Identify the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>Identify the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\r\n \t<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex] shifted to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\r\n \t<li>Sketch the asymptotes[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\r\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_06_02_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1700600\">\r\n<div id=\"fs-id1700602\">\r\n<h3>Graphing a Modified Cotangent<\/h3>\r\n<p id=\"fs-id1690330\">Sketch a graph of one period of the function[latex]\\,f\\left(x\\right)=4\\mathrm{cot}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"352430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"352430\"]\r\n<ul id=\"eip-id1165134038202\">\r\n \t<li><em>Step 1.<\/em> The function is already written in the general form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D.[\/latex]<\/li>\r\n \t<li><em>Step 2.<\/em>[latex]\\,A=4,[\/latex]so the stretching factor is 4.<\/li>\r\n \t<li><em>Step 3.<\/em>[latex]\\,B=\\frac{\\pi }{8},[\/latex]so the period is[latex]\\,P=\\frac{\\pi }{|B|}=\\frac{\\pi }{\\frac{\\pi }{8}}=8.[\/latex]<\/li>\r\n \t<li><em>Step 4.<\/em>[latex]\\,C=\\frac{\\pi }{2},[\/latex]so the phase shift is[latex]\\,\\frac{C}{B}=\\frac{\\frac{\\pi }{2}}{\\frac{\\pi }{8}}=4.[\/latex]<\/li>\r\n \t<li><em>Step 5.<\/em> We draw[latex]\\,f\\left(x\\right)=4\\mathrm{tan}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/li>\r\n \t<li><em>Step 6-7.<\/em> Three points we can use to guide the graph are[latex]\\,\\left(6,2\\right),\\left(8,-2\\right),\\,[\/latex]and[latex]\\,\\left(10,-6\\right).\\,[\/latex]We use the reciprocal relationship of tangent and cotangent to draw[latex]\\,f\\left(x\\right)=4\\mathrm{cot}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/li>\r\n \t<li><em>Step 8.<\/em> The vertical asymptotes are[latex]\\,x=4\\,[\/latex]and[latex]\\,x=12.[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1445913\">The graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_020\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_02_020\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143810\/CNX_Precalc_Figure_06_02_020.jpg\" alt=\"A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12.\" width=\"487\" height=\"315\" \/> <strong>Figure 15. <\/strong>One period of a modified cotangent function[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1461208\" class=\"bc-section section\">\r\n<h3>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h3>\r\n<p id=\"fs-id1370674\">Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the <span class=\"no-emphasis\">tangent function<\/span>.<\/p>\r\n\r\n<div id=\"Example_06_02_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1270016\">\r\n<div id=\"fs-id1270018\">\r\n<h3>Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\r\n<p id=\"fs-id1535059\">Suppose the function[latex]\\,y=5\\mathrm{tan}\\left(\\frac{\\pi }{4}t\\right)\\,[\/latex]marks the distance in the movement of a light beam from the top of a police car across a wall where[latex]\\,t\\,[\/latex]is the time in seconds and[latex]\\,y\\,[\/latex]is the distance in feet from a point on the wall directly across from the police car.<\/p>\r\n\r\n<ol id=\"fs-id1526727\" type=\"a\">\r\n \t<li>Find and interpret the stretching factor and period.<\/li>\r\n \t<li>Graph on the interval[latex]\\,\\left[0,5\\right].[\/latex]<\/li>\r\n \t<li>Evaluate[latex]\\,f\\left(1\\right)\\,[\/latex]and discuss the function\u2019s value at that input.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1658187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1658187\"]\r\n<ol id=\"fs-id1658187\" type=\"a\">\r\n \t<li>We know from the general form of[latex]\\,y=A\\mathrm{tan}\\left(Bt\\right)\\,[\/latex]that[latex]\\,|A|\\,[\/latex]is the stretching factor and[latex]\\,\\frac{\\pi }{B}\\,[\/latex]is the period.\r\n<div id=\"Image_06_02_022\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143823\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/> <strong>Figure 16.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id887356\">We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.<\/p>\r\n<p id=\"fs-id1651541\">The period is[latex]\\,\\frac{\\pi }{\\frac{\\pi }{4}}=\\frac{\\pi }{1}\\cdot \\frac{4}{\\pi }=4.\\,[\/latex]This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/p>\r\n<\/li>\r\n \t<li>To graph the function, we draw an asymptote at[latex]\\,t=2\\,[\/latex]and use the stretching factor and period. See <a class=\"autogenerated-content\" href=\"#Image_06_02_021\">(Figure)<\/a>\r\n<div id=\"Image_06_02_021\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143828\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/> <strong>Figure 17.<\/strong>[\/caption]\r\n\r\n<\/div><\/li>\r\n \t<li>period:[latex]\\,f\\left(1\\right)=5\\mathrm{tan}\\left(\\frac{\\pi }{4}\\left(1\\right)\\right)=5\\left(1\\right)=5;\\,[\/latex]after 1 second, the beam of has moved 5 ft from the spot across from the police car.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1701460\" class=\"precalculus media\">\r\n<p id=\"fs-id1516989\">Access these online resources for additional instruction and practice with graphs of other trigonometric functions.<\/p>\r\n\r\n<ul id=\"fs-id1526866\">\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphtangent\">Graphing the Tangent<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphcscsec\">Graphing Cosecant and Secant<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphcot\">Graphing the Cotangent<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1526416\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165133401597\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\r\n<td>[latex]y=A\\,\\mathrm{tan}\\left(Bx-C\\right)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\r\n<td>[latex]y=A\\,\\mathrm{sec}\\left(Bx-C\\right)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched cosecant function<\/td>\r\n<td>[latex]y=A\\,\\mathrm{csc}\\left(Bx-C\\right)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\r\n<td>[latex]y=A\\,\\mathrm{cot}\\left(Bx-C\\right)+D[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1693998\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1651799\">\r\n \t<li>The tangent function has period[latex]\\,\\pi .[\/latex]<\/li>\r\n \t<li>[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(Bx-C\\right)+D\\,[\/latex]is a tangent with vertical and\/or horizontal stretch\/compression and shift. See <a class=\"autogenerated-content\" href=\"#Example_06_02_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_06_02_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_06_02_03\">(Figure)<\/a>.<\/li>\r\n \t<li>The secant and cosecant are both periodic functions with a period of[latex]\\,2\\pi .\\,[\/latex][latex]f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]gives a shifted, compressed, and\/or stretched secant function graph. See <a class=\"autogenerated-content\" href=\"#Example_06_02_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_05\">(Figure)<\/a>.<\/li>\r\n \t<li>[latex]f\\left(x\\right)=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,[\/latex]gives a shifted, compressed, and\/or stretched cosecant function graph. See <a class=\"autogenerated-content\" href=\"#Example_06_02_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_07\">(Figure)<\/a>.<\/li>\r\n \t<li>The cotangent function has period[latex]\\,\\pi \\,[\/latex]and vertical asymptotes at[latex]\\,0,\u00b1\\pi ,\u00b12\\pi ,....[\/latex]<\/li>\r\n \t<li>The range of cotangent is[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]and the function is decreasing at each point in its range.<\/li>\r\n \t<li>The cotangent is zero at[latex]\\,\u00b1\\frac{\\pi }{2},\u00b1\\frac{3\\pi }{2},....[\/latex]<\/li>\r\n \t<li>[latex]f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D\\,[\/latex]is a cotangent with vertical and\/or horizontal stretch\/compression and shift. See <a class=\"autogenerated-content\" href=\"#Example_06_02_08\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_09\">(Figure)<\/a>.<\/li>\r\n \t<li>Real-world scenarios can be solved using graphs of trigonometric functions. See <a class=\"autogenerated-content\" href=\"#Example_06_02_10\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id878869\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id878873\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1407115\">\r\n<div id=\"fs-id1407118\">\r\n<p id=\"fs-id1407119\">Explain how the graph of the sine function can be used to graph[latex]\\,y=\\mathrm{csc}\\,x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1439607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1439607\"]\r\n<p id=\"fs-id1439607\">Since[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]is the reciprocal function of[latex]\\,y=\\mathrm{sin}\\,x,\\,[\/latex]you can plot the reciprocal of the coordinates on the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]to obtain the <em>y<\/em>-coordinates of[latex]\\,y=\\mathrm{csc}\\,x.\\,[\/latex]The <em>x<\/em>-intercepts of the graph[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]are the vertical asymptotes for the graph of[latex]\\,y=\\mathrm{csc}\\,x.[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1513360\">\r\n<div id=\"fs-id1513362\">\r\n<p id=\"fs-id1454682\">How can the graph of[latex]\\,y=\\mathrm{cos}\\,x\\,[\/latex]be used to construct the graph of[latex]\\,y=\\mathrm{sec}\\,x?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1421392\">\r\n<div id=\"fs-id1421394\">\r\n<p id=\"fs-id1421396\">Explain why the period of[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]is equal to[latex]\\,\\pi .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1461033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1461033\"]\r\n<p id=\"fs-id1461033\">Answers will vary. Using the unit circle, one can show that[latex]\\,\\mathrm{tan}\\left(x+\\pi \\right)=\\mathrm{tan}\\,x.\\,[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1409386\">\r\n<div id=\"fs-id1409388\">\r\n<p id=\"fs-id1351021\">Why are there no intercepts on the graph of[latex]\\,y=\\mathrm{csc}\\,x?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1529165\">\r\n<div id=\"fs-id1529167\">\r\n<p id=\"fs-id1529169\">How does the period of[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]compare with the period of[latex]\\,y=\\mathrm{sin}\\,x?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1365332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1365332\"]\r\n<p id=\"fs-id1365332\">The period is the same:[latex]\\,2\\pi .[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1608979\" class=\"bc-section section\">\r\n<h4>Algebraic<\/h4>\r\n<p id=\"fs-id1608984\">For the following exercises, match each trigonometric function with one of the following graphs.<\/p>\r\n<span id=\"Figure_06_02_201a\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143830\/CNX_Precalc_Figure_06_02_201a.jpg\" alt=\"Trigonometric graph of tangent of x.\" \/><\/span><span id=\"Figure_06_02_201b\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143832\/CNX_Precalc_Figure_06_02_201b.jpg\" alt=\"Trigonometric graph of secant of x.\" \/><\/span><span id=\"Figure_06_02_201c\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143837\/CNX_Precalc_Figure_06_02_201c.jpg\" alt=\"Trigonometric graph of cosecant of x.\" \/><\/span><span id=\"Figure_06_02_201d\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143844\/CNX_Precalc_Figure_06_02_201d.jpg\" alt=\"Trigonometric graph of cotangent of x.\" \/><\/span>\r\n<div id=\"fs-id1600438\">\r\n<div id=\"fs-id1600440\">\r\n<p id=\"fs-id1600443\">[latex]f\\left(x\\right)=\\mathrm{tan}\\,x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1402861\">\r\n<div id=\"fs-id1402863\">\r\n<p id=\"fs-id1402865\">[latex]f\\left(x\\right)=\\mathrm{sec}\\,x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1512360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1512360\"]\r\n<p id=\"fs-id1512360\">IV<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1512365\">\r\n<div id=\"fs-id1451983\">\r\n<p id=\"fs-id1451985\">[latex]f\\left(x\\right)=\\mathrm{csc}\\,x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1434924\">\r\n<div id=\"fs-id1434926\">\r\n<p id=\"fs-id1673834\">[latex]f\\left(x\\right)=\\mathrm{cot}\\,x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1410903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1410903\"]\r\n<p id=\"fs-id1410903\">III<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1365640\">For the following exercises, find the period and horizontal shift of each of the functions.<\/p>\r\n\r\n<div id=\"fs-id1365643\">\r\n<div id=\"fs-id1365645\">\r\n<p id=\"fs-id1562031\">[latex]f\\left(x\\right)=2\\mathrm{tan}\\left(4x-32\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1662387\">\r\n<div id=\"fs-id1662389\">\r\n<p id=\"fs-id1671047\">[latex]h\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{4}\\left(x+1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1581685\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1581685\"]\r\n<p id=\"fs-id1581685\">period: 8; horizontal shift: 1 unit to left<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1405674\">\r\n<div id=\"fs-id1405676\">\r\n<p id=\"fs-id1405678\">[latex]m\\left(x\\right)=6\\mathrm{csc}\\left(\\frac{\\pi }{3}x+\\pi \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1652255\">\r\n<div id=\"fs-id1652257\">\r\n<p id=\"fs-id1652259\">If[latex]\\,\\mathrm{tan}\\,x=-1.5,\\,[\/latex]find[latex]\\,\\mathrm{tan}\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1429264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1429264\"]\r\n<p id=\"fs-id1429264\">1.5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1422443\">\r\n<div id=\"fs-id1422445\">\r\n<p id=\"fs-id1422447\">If[latex]\\,\\mathrm{sec}\\,x=2,\\,[\/latex]find[latex]\\,\\mathrm{sec}\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1356505\">\r\n<div id=\"fs-id1456034\">\r\n<p id=\"fs-id1456037\">If[latex]\\,\\mathrm{csc}\\,x=-5,\\,[\/latex]find[latex]\\,\\mathrm{csc}\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1375607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1375607\"]\r\n<p id=\"fs-id1375607\">5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1404272\">\r\n<div id=\"fs-id1404274\">\r\n<p id=\"fs-id1428797\">If[latex]\\,x\\mathrm{sin}\\,x=2,\\,[\/latex]find[latex]\\,\\left(-x\\right)\\mathrm{sin}\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1508271\">For the following exercises, rewrite each expression such that the argument[latex]\\,x\\,[\/latex]is positive.<\/p>\r\n\r\n<div id=\"fs-id1351072\">\r\n<div id=\"fs-id1351074\">\r\n<p id=\"fs-id1351076\">[latex]\\mathrm{cot}\\left(-x\\right)\\mathrm{cos}\\left(-x\\right)+\\mathrm{sin}\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1407073\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1407073\"]\r\n<p id=\"fs-id1407073\">[latex]-\\mathrm{cot}x\\mathrm{cos}x-\\mathrm{sin}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1458413\">\r\n<div id=\"fs-id1458415\">\r\n<p id=\"fs-id1458417\">[latex]\\mathrm{cos}\\left(-x\\right)+\\mathrm{tan}\\left(-x\\right)\\mathrm{sin}\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1513346\" class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1283129\">For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.<\/p>\r\n\r\n<div id=\"fs-id1283134\">\r\n<div id=\"fs-id1434575\">\r\n<p id=\"fs-id1434577\">[latex]f\\left(x\\right)=2\\mathrm{tan}\\left(4x-32\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"46974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"46974\"]\r\n\r\n<span id=\"fs-id1644075\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143846\/CNX_Precalc_Figure_06_02_202.jpg\" alt=\"A graph of two periods of a modified tangent function. There are two vertical asymptotes.\" \/><\/span>\r\n<p id=\"fs-id1656875\">stretching factor: 2; period:[latex]\\text{ }\\frac{\\pi }{4};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{1}{4}\\left(\\frac{\\pi }{2}+\\pi k\\right)+8,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1656875\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div id=\"fs-id1422235\">\r\n<p id=\"fs-id1422238\">[latex]\\,h\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{4}\\left(x+1\\right)\\right)\\,[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1435841\">\r\n<div id=\"fs-id1419768\">\r\n<p id=\"fs-id1419770\">[latex]m\\left(x\\right)=6\\mathrm{csc}\\left(\\frac{\\pi }{3}x+\\pi \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"278253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278253\"]\r\n\r\n<span id=\"fs-id1375745\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143851\/CNX_Precalc_Figure_06_02_204.jpg\" alt=\"A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.\" \/><\/span>\r\n<p id=\"fs-id1512258\">stretching factor: 6; period: 6; asymptotes:[latex]\\text{ }x=3k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1512258\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1562828\">\r\n<div id=\"fs-id1562830\">\r\n<p id=\"fs-id1562832\">[latex]j\\left(x\\right)=\\mathrm{tan}\\left(\\frac{\\pi }{2}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1581814\">\r\n<div id=\"fs-id1581816\">\r\n<p id=\"fs-id1581818\">[latex]p\\left(x\\right)=\\mathrm{tan}\\left(x-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"528553\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"528553\"]\r\n\r\n<span id=\"fs-id1365984\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143854\/CNX_Precalc_Figure_06_02_206.jpg\" alt=\"A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.\" \/><\/span>\r\n<p id=\"fs-id1419808\">stretching factor: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1419808\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1385170\">\r\n<div id=\"fs-id1385172\">\r\n<p id=\"fs-id1385174\">[latex]f\\left(x\\right)=4\\mathrm{tan}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1506812\">\r\n<div id=\"fs-id1506814\">\r\n<p id=\"fs-id1506816\">[latex]f\\left(x\\right)=\\mathrm{tan}\\left(x+\\frac{\\pi }{4}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"156444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"156444\"]\r\n\r\n<span id=\"fs-id1354983\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143857\/CNX_Precalc_Figure_06_02_208.jpg\" alt=\"A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.\" \/><\/span>\r\n<p id=\"fs-id1528080\">Stretching factor: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1528080\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1364876\">\r\n<div id=\"fs-id1364878\">\r\n<p id=\"fs-id1364880\">[latex]f\\left(x\\right)=\\pi \\mathrm{tan}\\left(\\pi x-\\pi \\right)-\\pi [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1506136\">\r\n<div id=\"fs-id1655270\">\r\n<p id=\"fs-id1655272\">[latex]f\\left(x\\right)=2\\mathrm{csc}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"970468\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"970468\"]\r\n\r\n<span id=\"fs-id1455973\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143903\/CNX_Precalc_Figure_06_02_210.jpg\" alt=\"A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.\" \/><\/span>\r\n<p id=\"fs-id1489928\">stretching factor: 2; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1489928\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1402973\">\r\n<div id=\"fs-id1402975\">\r\n<p id=\"fs-id1405719\">[latex]f\\left(x\\right)=-\\frac{1}{4}\\mathrm{csc}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1699640\">\r\n<div id=\"fs-id1408399\">\r\n<p id=\"fs-id1408401\">[latex]f\\left(x\\right)=4\\mathrm{sec}\\left(3x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"999274\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"999274\"]\r\n\r\n<span id=\"fs-id1704371\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143909\/CNX_Precalc_Figure_06_02_212.jpg\" alt=\"A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi\/2, -pi\/6, pi\/6, and pi\/2.\" \/><\/span>\r\n<p id=\"fs-id1583862\">stretching factor: 4; period:[latex]\\text{ }\\frac{2\\pi }{3};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{6}k,\\text{ where }k\\text{ is an odd integer}[\/latex]<\/p>\r\n<p id=\"fs-id1583862\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1705843\">\r\n<div id=\"fs-id1705846\">\r\n<p id=\"fs-id1705848\">[latex]f\\left(x\\right)=-3\\mathrm{cot}\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1375944\">\r\n<div id=\"fs-id1420097\">\r\n<p id=\"fs-id1420099\">[latex]f\\left(x\\right)=7\\mathrm{sec}\\left(5x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"643578\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"643578\"]\r\n\r\n<span id=\"fs-id1429978\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143916\/CNX_Precalc_Figure_06_02_214.jpg\" alt=\"A graph of two periods of a modified secant function. There are four vertical asymptotes all pi\/5 apart.\" \/><\/span>\r\n<p id=\"fs-id1582375\">stretching factor: 7; period:[latex]\\text{ }\\frac{2\\pi }{5};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{10}k,\\text{ where }k\\text{ is an odd integer}[\/latex]<\/p>\r\n<p id=\"fs-id1582375\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1655075\">\r\n<div id=\"fs-id1655077\">\r\n<p id=\"fs-id1417308\">[latex]f\\left(x\\right)=\\frac{9}{10}\\mathrm{csc}\\left(\\pi x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1523811\">\r\n<div id=\"fs-id1523813\">\r\n<p id=\"fs-id1523815\">[latex]f\\left(x\\right)=2\\mathrm{csc}\\left(x+\\frac{\\pi }{4}\\right)-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"125458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"125458\"]\r\n\r\n<span id=\"fs-id1404603\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143918\/CNX_Precalc_Figure_06_02_216.jpg\" alt=\"A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.\" \/><\/span>\r\n<p id=\"fs-id1361638\">stretching factor: 2; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=-\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1361638\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1434375\">\r\n<div id=\"fs-id1434377\">\r\n<p id=\"fs-id1434379\">[latex]f\\left(x\\right)=-\\mathrm{sec}\\left(x-\\frac{\\pi }{3}\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1614813\">\r\n<div id=\"fs-id1614815\">\r\n<p id=\"fs-id1614818\">[latex]f\\left(x\\right)=\\frac{7}{5}\\mathrm{csc}\\left(x-\\frac{\\pi }{4}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n[reveal-answer q=\"538264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"538264\"]\r\n\r\n<span id=\"fs-id1446559\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143921\/CNX_Precalc_Figure_06_02_218.jpg\" alt=\"A graph of a modified cosecant function. Four vertical asymptotes.\" \/><\/span>\r\n<p id=\"fs-id1375388\">stretching factor:[latex]\\text{ }\\frac{7}{5};\\text{ }[\/latex]period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\r\n<p id=\"fs-id1375388\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1409971\">\r\n<div id=\"fs-id1409974\">\r\n<p id=\"fs-id1409976\">[latex]f\\left(x\\right)=5\\left(\\mathrm{cot}\\left(x+\\frac{\\pi }{2}\\right)-3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1464121\">For the following exercises, find and graph two periods of the periodic function with the given stretching factor,[latex]\\,|A|,\\,[\/latex]period, and phase shift.<\/p>\r\n\r\n<div>\r\n<div>\r\n<p id=\"fs-id1649334\">A tangent curve,[latex]\\,A=1,\\,[\/latex]period of[latex]\\,\\frac{\\pi }{3};\\,[\/latex]and phase shift[latex]\\,\\left(h,\\,k\\right)=\\left(\\frac{\\pi }{4},2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1694857\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1694857\"]\r\n<p id=\"fs-id1694857\">[latex]y=\\mathrm{tan}\\left(3\\left(x-\\frac{\\pi }{4}\\right)\\right)+2[\/latex]<\/p>\r\n<span id=\"fs-id1649707\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143924\/CNX_Precalc_Figure_06_02_220.jpg\" alt=\"A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi\/4 and pi\/12.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1658055\">\r\n<div id=\"fs-id1440828\">\r\n<p id=\"fs-id1440830\">A tangent curve,[latex]\\,A=-2,\\,[\/latex]period of[latex]\\,\\frac{\\pi }{4},\\,[\/latex]and phase shift[latex]\\,\\left(h,\\,k\\right)=\\left(-\\frac{\\pi }{4},\\,-2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1588386\">For the following exercises, find an equation for the graph of each function.<\/p>\r\n\r\n<div id=\"fs-id1588389\">\r\n<div id=\"fs-id1588390\"><span id=\"fs-id1513948\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143931\/CNX_Precalc_Figure_06_02_222.jpg\" alt=\"A graph of two periods of a modified cosecant function, with asymptotes at multiples of pi\/2.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1706414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1706414\"]\r\n<p id=\"fs-id1706414\">[latex]f\\left(x\\right)=\\mathrm{csc}\\left(2x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1464236\">\r\n<div id=\"fs-id1464237\"><span id=\"fs-id1537949\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143933\/CNX_Precalc_Figure_06_02_223.jpg\" alt=\"A graph of a modified cotangent function. Vertical asymptotes at x=-1 and x=0 and x=1.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1409010\">\r\n<div id=\"fs-id1409012\"><span id=\"fs-id1409017\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143936\/CNX_Precalc_Figure_06_02_224.jpg\" alt=\"A graph of a modified cosecant function. Vertical asymptotes at multiples of pi\/4.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1405484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1405484\"]\r\n<p id=\"fs-id1405484\">[latex]f\\left(x\\right)=\\mathrm{csc}\\left(4x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1465129\">\r\n<div id=\"fs-id1465130\"><span id=\"fs-id1465137\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143949\/CNX_Precalc_Figure_06_02_225.jpg\" alt=\"A graph of a modified tangent function. Vertical asymptotes at -pi\/8 and 3pi\/8.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1653766\">\r\n<div id=\"fs-id1653767\"><span id=\"fs-id1445558\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143954\/CNX_Precalc_Figure_06_02_226.jpg\" alt=\"A graph of a modified cosecant function. Vertical asymptotyes at multiples of pi.\" \/><\/span><\/div>\r\n<div id=\"fs-id705620\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id705620\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id705620\"]\r\n<p id=\"fs-id705621\">[latex]f\\left(x\\right)=2\\mathrm{csc}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1530295\">\r\n<div id=\"fs-id1530296\"><span id=\"fs-id1695827\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143957\/CNX_Precalc_Figure_06_02_227.jpg\" alt=\"A graph of a modified secant function. Four vertical asymptotes.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1664599\">\r\n<div id=\"fs-id1664600\"><span id=\"fs-id1664606\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143959\/CNX_Precalc_Figure_06_02_228.jpg\" alt=\"graph of two periods of a modified tangent function. Vertical asymptotes at x=-0.005 and x=0.005.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1460872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1460872\"]\r\n<p id=\"fs-id1460872\">[latex]f\\left(x\\right)=\\frac{1}{2}\\mathrm{tan}\\left(100\\pi x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1678301\" class=\"bc-section section\">\r\n<h4>Technology<\/h4>\r\n<p id=\"fs-id1678306\">For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input[latex]\\,\\mathrm{csc}\\,x\\,[\/latex]as[latex]\\,\\frac{1}{\\mathrm{sin}\\,x}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1691184\">\r\n<div id=\"fs-id1691186\">\r\n<p id=\"fs-id1691188\">[latex]f\\left(x\\right)=|\\mathrm{csc}\\left(x\\right)|[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1690901\">\r\n<div id=\"fs-id1447025\">\r\n<p id=\"fs-id1447027\">[latex]f\\left(x\\right)=|\\mathrm{cot}\\left(x\\right)|[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1656302\"]Show Solution[\/reveal-answer][hidden-answer a=\"1656302\"]<span id=\"fs-id1656308\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144004\/CNX_Precalc_Figure_06_02_230.jpg\" alt=\"A graph of the absolute value of the cotangent function. Range is 0 to infinity.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1699978\">\r\n<div id=\"fs-id1699980\">\r\n<p id=\"fs-id1699982\">[latex]f\\left(x\\right)={2}^{\\mathrm{csc}\\left(x\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<p id=\"fs-id1451630\">[latex]f\\left(x\\right)=\\frac{\\mathrm{csc}\\left(x\\right)}{\\mathrm{sec}\\left(x\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1420323\"]Show Solution[\/reveal-answer][hidden-answer a=\"1420323\"]<span id=\"fs-id1420330\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144011\/CNX_Precalc_Figure_06_02_232.jpg\" alt=\"A graph of tangent of x.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1521332\">\r\n<div id=\"fs-id1586756\">\r\n<p id=\"fs-id1586758\">Graph[latex]\\,f\\left(x\\right)=1+{\\mathrm{sec}}^{2}\\left(x\\right)-{\\mathrm{tan}}^{2}\\left(x\\right).\\,[\/latex]What is the function shown in the graph?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1600902\">\r\n<div id=\"fs-id1600904\">\r\n<p id=\"fs-id1600906\">[latex]f\\left(x\\right)=\\mathrm{sec}\\left(0.001x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1586275\"]Show Solution[\/reveal-answer][hidden-answer a=\"1586275\"]<span id=\"fs-id1586282\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144019\/CNX_Precalc_Figure_06_02_234.jpg\" alt=\"A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1432442\">\r\n<div id=\"fs-id1432444\">\r\n<p id=\"fs-id1671603\">[latex]f\\left(x\\right)=\\mathrm{cot}\\left(100\\pi x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1536091\">\r\n<div id=\"fs-id1536094\">\r\n<p id=\"fs-id1536096\">[latex]f\\left(x\\right)={\\mathrm{sin}}^{2}x+{\\mathrm{cos}}^{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"1631004\"]Show Solution[\/reveal-answer][hidden-answer a=\"1631004\"]<span id=\"fs-id1631010\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144024\/CNX_Precalc_Figure_06_02_241.jpg\" alt=\"A graph of y=1.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1526988\" class=\"bc-section section\">\r\n<h4>Real-World Applications<\/h4>\r\n<div id=\"fs-id1526993\">\r\n<div id=\"fs-id1526995\">\r\n<p id=\"fs-id1405403\">The function[latex]\\,f\\left(x\\right)=20\\mathrm{tan}\\left(\\frac{\\pi }{10}x\\right)\\,[\/latex]marks the distance in the movement of a light beam from a police car across a wall for time[latex]\\,x,\\,[\/latex]in seconds, and distance[latex]\\,f\\left(x\\right),[\/latex]\r\nin feet.<\/p>\r\n\r\n<ol id=\"fs-id1384936\" type=\"a\">\r\n \t<li>Graph on the interval[latex]\\,\\left[0,\\,5\\right].[\/latex]<\/li>\r\n \t<li>Find and interpret the stretching factor, period, and asymptote.<\/li>\r\n \t<li>Evaluate[latex]\\,f\\left(1\\right)\\,[\/latex]and[latex]\\,f\\left(2.5\\right)\\,[\/latex]and discuss the function\u2019s values at those inputs.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1513694\">\r\n<div id=\"fs-id1513696\">\r\n<p id=\"fs-id1513698\">Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let[latex]\\,x,\\,[\/latex]measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and[latex]\\,x\\,[\/latex]is measured negative to the left and positive to the right. (See <a class=\"autogenerated-content\" href=\"#Figure_06_02_237\">(Figure)<\/a>.) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance[latex]\\,d\\left(x\\right),\\,[\/latex]in kilometers, from the fisherman to the boat is given by the function[latex]\\,d\\left(x\\right)=1.5\\mathrm{sec}\\left(x\\right).[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1434023\" type=\"a\">\r\n \t<li>What is a reasonable domain for[latex]\\,d\\left(x\\right)?[\/latex]<\/li>\r\n \t<li>Graph[latex]\\,d\\left(x\\right)\\,[\/latex]on this domain.<\/li>\r\n \t<li>Find and discuss the meaning of any vertical asymptotes on the graph of[latex]\\,d\\left(x\\right).[\/latex]<\/li>\r\n \t<li>Calculate and interpret[latex]\\,d\\left(-\\frac{\\pi }{3}\\right).\\,[\/latex]Round to the second decimal place.<\/li>\r\n \t<li>Calculate and interpret[latex]\\,d\\left(\\frac{\\pi }{6}\\right).\\,[\/latex]Round to the second decimal place.<\/li>\r\n \t<li>What is the minimum distance between the fisherman and the boat? When does this occur?<\/li>\r\n<\/ol>\r\n<div id=\"Figure_06_02_237\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"465\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144035\/CNX_Precalc_Figure_06_02_237.jpg\" alt=\"An illustration of a man and the distance he is away from a boat.\" width=\"465\" height=\"280\" \/> <strong>Figure 18.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1700717\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1700717\"]\r\n<ol id=\"fs-id1700717\" type=\"a\">\r\n \t<li>[latex]\\,\\left(-\\frac{\\pi }{2},\\,\\frac{\\pi }{2}\\right);\\,[\/latex]<\/li>\r\n \t<li><span id=\"fs-id1532106\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144038\/CNX_Precalc_Figure_06_02_238.jpg\" alt=\"A graph of a half period of a secant function. Vertical asymptotes at x=-pi\/2 and pi\/2.\" \/><\/span><\/li>\r\n \t<li>[latex]\\,x=-\\frac{\\pi }{2}\\,[\/latex]and[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]the distance grows without bound as[latex]\\,|x|[\/latex]approaches[latex]\\,\\frac{\\pi }{2}\\,[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;<\/li>\r\n \t<li>3; when[latex]\\,x=-\\frac{\\pi }{3},\\,[\/latex]the boat is 3 km away;<\/li>\r\n \t<li>1.73; when[latex]\\,x=\\frac{\\pi }{6},\\,[\/latex]the boat is about 1.73 km away;<\/li>\r\n \t<li>1.5 km; when[latex]\\,x=0\\,[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1431336\">\r\n<div id=\"fs-id1431338\">\r\n<p id=\"fs-id1431340\">A laser rangefinder is locked on a comet approaching Earth. The distance[latex]\\,g\\left(x\\right),\\,[\/latex]in kilometers, of the comet after[latex]\\,x\\,[\/latex]days, for[latex]\\,x\\,[\/latex]in the interval 0 to 30 days, is given by[latex]\\,g\\left(x\\right)=250,000\\mathrm{csc}\\left(\\frac{\\pi }{30}x\\right).[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1526530\" type=\"a\">\r\n \t<li>Graph[latex]\\,g\\left(x\\right)\\,[\/latex]on the interval[latex]\\,\\left[0,\\,35\\right].[\/latex]<\/li>\r\n \t<li>Evaluate[latex]\\,g\\left(5\\right)\\,[\/latex]\r\nand interpret the information.<\/li>\r\n \t<li>What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?<\/li>\r\n \t<li>Find and discuss the meaning of any vertical asymptotes.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1601867\">\r\n<div id=\"fs-id1601869\">\r\n<p id=\"fs-id1422576\">A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after[latex]\\,x\\,[\/latex]seconds is[latex]\\,\\frac{\\pi }{120}x.[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1614554\" type=\"a\">\r\n \t<li>Write a function expressing the altitude[latex]\\,h\\left(x\\right),\\,[\/latex]in miles, of the rocket above the ground after[latex]\\,x\\,[\/latex]seconds. Ignore the curvature of the Earth.<\/li>\r\n \t<li>Graph[latex]\\,h\\left(x\\right)\\,[\/latex]on the interval[latex]\\,\\left(0,\\,60\\right).[\/latex]<\/li>\r\n \t<li>Evaluate and interpret the values[latex]\\,h\\left(0\\right)\\,[\/latex]and[latex]\\,h\\left(30\\right).[\/latex]<\/li>\r\n \t<li>What happens to the values of[latex]\\,h\\left(x\\right)\\,[\/latex]as [latex]\\,x\\,[\/latex]\r\napproaches 60 seconds? Interpret the meaning of this in terms of the problem.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1523906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1523906\"]\r\n<ol id=\"fs-id1523906\" type=\"a\">\r\n \t<li>[latex]h\\left(x\\right)=2\\mathrm{tan}\\left(\\frac{\\pi }{120}x\\right);[\/latex]<\/li>\r\n \t<li><span id=\"fs-id1601972\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144041\/CNX_Precalc_Figure_06_02_240.jpg\" alt=\"An exponentially increasing function with a vertical asymptote at x=60.\" \/><\/span><\/li>\r\n \t<li>[latex]h\\left(0\\right)=0:\\,[\/latex]after 0 seconds, the rocket is 0 mi above the ground;[latex]h\\left(30\\right)=2:\\,[\/latex]after 30 seconds, the rockets is 2 mi high;<\/li>\r\n \t<li>As[latex]\\,x\\,[\/latex]approaches 60 seconds, the values of[latex]\\,h\\left(x\\right)\\,[\/latex]grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Analyze the graph of \u2009y=tan\u2009x.<\/li>\n<li>Graph variations of \u2009y=tan\u2009x.<\/li>\n<li>Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li>Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li>Analyze the graph of \u2009y=cot\u2009x.<\/li>\n<li>Graph variations of \u2009y=cot\u2009x.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1420477\">We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.<\/p>\n<div id=\"fs-id1321556\" class=\"bc-section section\">\n<h3>Analyzing the Graph of <em>y<\/em> = tan <em>x<\/em><\/h3>\n<p id=\"fs-id1530665\">We will begin with the graph of the <span class=\"no-emphasis\">tangent<\/span> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\n<div id=\"fs-id1614888\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\,x=\\frac{\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}[\/latex]<\/div>\n<p id=\"fs-id1288893\">The <span class=\"no-emphasis\">period<\/span> of the tangent function is[latex]\\,\\pi \\,[\/latex]because the graph repeats itself on intervals of[latex]\\,k\\pi \\,[\/latex]where[latex]\\,k\\,[\/latex]is a constant. If we graph the tangent function on[latex]\\,-\\frac{\\pi }{2}\\,[\/latex]to[latex]\\,\\frac{\\pi }{2},\\,[\/latex]we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.<\/p>\n<p id=\"fs-id1647294\">We can determine whether tangent is an odd or even function by using the definition of tangent.<\/p>\n<div id=\"fs-id1615402\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\mathrm{tan}\\left(-x\\right)=\\frac{\\mathrm{sin}\\left(-x\\right)}{\\mathrm{cos}\\left(-x\\right)}\\hfill & \\begin{array}{ccc}& & \\end{array}\\text{Definition of tangent}.\\hfill \\\\ \\text{ }=\\frac{-\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}\\hfill & \\begin{array}{ccc}& & \\end{array}\\text{Sine is an odd function, cosine is even}.\\hfill \\\\ \\text{ }=-\\frac{\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}\\hfill & \\begin{array}{ccc}& & \\end{array}\\text{The quotient of an odd and an even function is odd}.\\hfill \\\\ \\text{ }=-\\mathrm{tan}\\,x\\hfill & \\begin{array}{ccc}& & \\end{array}\\text{Definition of tangent}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1460636\">Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in <a class=\"autogenerated-content\" href=\"#Table_06_02_00\">(Figure)<\/a>.<\/p>\n<table id=\"Table_06_02_00\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]-\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]-\\frac{\\pi }{6}[\/latex]<\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>undefined<\/td>\n<td>[latex]-\\sqrt{3}[\/latex]<\/td>\n<td>\u20131<\/td>\n<td>[latex]-\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1279869\">These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when[latex]\\,\\frac{\\pi }{3}<x<\\frac{\\pi }{2},\\,[\/latex]we can use a table to look for a trend. Because[latex]\\,\\frac{\\pi }{3}\\approx 1.05\\,[\/latex]and[latex]\\,\\frac{\\pi }{2}\\approx 1.57,\\,[\/latex]we will evaluate[latex]\\,x\\,[\/latex]at radian measures[latex]\\,1.05<x<1.57\\,[\/latex]as shown in <a class=\"autogenerated-content\" href=\"#Table_06_02_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>1.3<\/td>\n<td>1.5<\/td>\n<td>1.55<\/td>\n<td>1.56<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\mathrm{tan} \\text{ }x[\/latex]<\/strong><\/td>\n<td>3.6<\/td>\n<td>14.1<\/td>\n<td>48.1<\/td>\n<td>92.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1658883\">As[latex]\\,x\\,[\/latex]approaches[latex]\\,\\frac{\\pi }{2},\\,[\/latex]the outputs of the function get larger and larger. Because[latex]\\,y=\\mathrm{tan}\\,x\\,[\/latex]is an odd function, we see the corresponding table of negative values in <a class=\"autogenerated-content\" href=\"#Table_06_02_02\">(Figure)<\/a>.<\/p>\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22121.3<\/td>\n<td>\u22121.5<\/td>\n<td>\u22121.55<\/td>\n<td>\u22121.56<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\mathrm{tan}\\,x[\/latex]<\/strong><\/td>\n<td>\u22123.6<\/td>\n<td>\u221214.1<\/td>\n<td>\u221248.1<\/td>\n<td>\u221292.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1318044\">We can see that, as[latex]\\,x\\,[\/latex]approaches[latex]\\,-\\frac{\\pi }{2},\\,[\/latex]the outputs get smaller and smaller. Remember that there are some values of[latex]\\,x\\,[\/latex]for which[latex]\\,\\mathrm{cos}\\,x=0.\\,[\/latex]For example,[latex]\\,\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0\\,[\/latex]and[latex]\\,\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0.\\,[\/latex]At these values, the <span class=\"no-emphasis\">tangent function<\/span> is undefined, so the graph of[latex]\\,y=\\mathrm{tan}\\,x\\,[\/latex]has discontinuities at[latex]\\,x=\\frac{\\pi }{2}\\text{ and }\\frac{3\\pi }{2}.\\,[\/latex]At these values, the graph of the tangent has vertical asymptotes. <a class=\"autogenerated-content\" href=\"#Figure_06_02_001\">(Figure)<\/a> represents the graph of[latex]\\,y=\\mathrm{tan}\\,x.\\,[\/latex]The tangent is positive from 0 to[latex]\\,\\frac{\\pi }{2}\\,[\/latex]and from[latex]\\,\\pi \\,[\/latex]to[latex]\\,\\frac{3\\pi }{2},\\,[\/latex]corresponding to quadrants I and III of the unit circle.<\/p>\n<div id=\"Figure_06_02_001\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143550\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Graph of the tangent function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1678025\" class=\"bc-section section\">\n<h3>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h3>\n<p id=\"fs-id1232225\">As with the sine and cosine functions, the <span class=\"no-emphasis\">tangent<\/span> function can be described by a general equation.<\/p>\n<div id=\"fs-id988668\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{tan}\\left(Bx\\right)[\/latex]<\/div>\n<p id=\"fs-id1393080\">We can identify horizontal and vertical stretches and compressions using values of[latex]\\,A\\,[\/latex]and[latex]\\,B.\\,[\/latex]The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.<\/p>\n<p id=\"fs-id933588\">Because there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant[latex]\\,A.[\/latex]<\/p>\n<div id=\"fs-id1409849\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\n<ul id=\"fs-id1300606\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is all real numbers[latex]\\,x,[\/latex]where[latex]\\,x\\ne \\frac{\\pi }{2|B|}+\\frac{\\pi }{|B|}k\\,[\/latex]such that[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\n<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{2|B|}+\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>[latex]y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex]is an odd function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1430247\" class=\"bc-section section\">\n<h4>Graphing One Period of a Stretched or Compressed Tangent Function<\/h4>\n<p id=\"fs-id1698843\">We can use what we know about the properties of the <span class=\"no-emphasis\">tangent function<\/span> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(Bx\\right).\\,[\/latex]We focus on a single <span class=\"no-emphasis\">period<\/span> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval[latex]\\,\\left(-\\frac{P}{2},\\frac{P}{2}\\right)\\,[\/latex]and the graph has vertical asymptotes at[latex]\\,\u00b1\\frac{P}{2}\\,[\/latex]where[latex]\\,P=\\frac{\\pi }{B}.\\,[\/latex]On[latex]\\,\\left(-\\frac{\\pi }{2},\\frac{\\pi }{2}\\right),\\,[\/latex]the graph will come up from the left asymptote at[latex]\\,x=-\\frac{\\pi }{2},\\,[\/latex]cross through the origin, and continue to increase as it approaches the right asymptote at[latex]\\,x=\\frac{\\pi }{2}.\\,[\/latex]To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use<\/p>\n<div id=\"fs-id1631239\" class=\"unnumbered aligncenter\">[latex]f\\left(\\frac{P}{4}\\right)=A\\mathrm{tan}\\left(B\\frac{P}{4}\\right)=A\\mathrm{tan}\\left(B\\frac{\\pi }{4B}\\right)=A[\/latex]<\/div>\n<p id=\"fs-id1638091\">because[latex]\\,\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1.[\/latex]<\/p>\n<div id=\"fs-id1644376\" class=\"precalculus howto\">\n<p id=\"fs-id917405\"><strong>Given the function[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"fs-id1270463\" type=\"1\">\n<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>Draw vertical asymptotes at[latex]\\,x=-\\frac{P}{2}\\,[\/latex]and[latex]\\,x=\\frac{P}{2}.[\/latex]<\/li>\n<li>For[latex]\\,A>0,\\,[\/latex]the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for[latex]\\,A<0[\/latex]).<\/li>\n<li>Plot reference points at[latex]\\,\\left(\\frac{P}{4},A\\right),\\,[\/latex][latex]\\left(0,0\\right),\\,[\/latex]and[latex]\\,\\left(-\\frac{P}{4},-A\\right),\\,[\/latex]and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_01\" class=\"textbox examples\">\n<div id=\"fs-id1426111\">\n<div id=\"fs-id1678052\">\n<h3>Sketching a Compressed Tangent<\/h3>\n<p id=\"fs-id1316312\">Sketch a graph of one period of the function[latex]\\,y=0.5\\mathrm{tan}\\left(\\frac{\\pi }{2}x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1051354\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1051354\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1051354\">First, we identify[latex]\\,A\\,[\/latex]and[latex]\\,B.[\/latex]<\/p>\n<p><span id=\"fs-id1654424\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143600\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" \/><\/span><\/p>\n<p id=\"fs-id1701807\">Because[latex]\\,A=0.5\\,[\/latex]and[latex]\\,B=\\frac{\\pi }{2},\\,[\/latex]we can find the <span class=\"no-emphasis\">stretching\/compressing factor<\/span> and period. The period is[latex]\\,\\frac{\\pi }{\\frac{\\pi }{2}}=2,\\,[\/latex]so the asymptotes are at[latex]\\,x=\u00b11.\\,[\/latex]At a quarter period from the origin, we have<\/p>\n<div id=\"fs-id1061460\">[latex]\\begin{array}{l}f\\left(0.5\\right)=0.5\\mathrm{tan}\\left(\\frac{0.5\\pi }{2}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.5\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0.5\\hfill \\end{array}[\/latex]<\/div>\n<p>This means the curve must pass through the points[latex]\\,\\left(0.5,0.5\\right),[\/latex][latex]\\left(0,0\\right),[\/latex]and[latex]\\,\\left(-0.5,-0.5\\right).\\,[\/latex]The only inflection point is at the origin. <a class=\"autogenerated-content\" href=\"#Figure_06_02_003\">(Figure)<\/a> shows the graph of one period of the function.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143606\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1009376\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_02\">\n<div id=\"fs-id1706988\">\n<p id=\"fs-id1706989\">Sketch a graph of[latex]\\,f\\left(x\\right)=3\\mathrm{tan}\\left(\\frac{\\pi }{6}x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1354903\">Show Solution<\/span><\/p>\n<div id=\"q1354903\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1439999\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143619\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1464435\" class=\"bc-section section\">\n<h4>Graphing One Period of a Shifted Tangent Function<\/h4>\n<p id=\"fs-id1406087\">Now that we can graph a <span class=\"no-emphasis\">tangent function<\/span> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add[latex]\\,C\\,[\/latex]and[latex]\\,D\\,[\/latex]to the general form of the tangent function.<\/p>\n<div id=\"fs-id1270990\">[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<p id=\"fs-id1241838\">The graph of a transformed tangent function is different from the basic tangent function[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]in several ways:<\/p>\n<div id=\"eip-87\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\n<ul id=\"fs-id1346106\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\n<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\,\\mathrm{tan}\\left(Bx-C\\right)+D\\,[\/latex]is an odd function because it is the quotient of odd and even functions (sin and cosine respectively).<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1665485\" class=\"precalculus howto\">\n<p id=\"fs-id1432077\"><strong>Given the function[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D,\\,[\/latex]sketch the graph of one period.<\/strong><\/p>\n<ol id=\"fs-id1459436\" type=\"1\">\n<li>Express the function given in the form[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li>Identify the <span class=\"no-emphasis\">stretching\/compressing factor<\/span>,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex]shifted to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\n<li>Sketch the vertical asymptotes, which occur at[latex]\\text{ }x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\text{ }k\\text{ }[\/latex]is an odd integer.<\/li>\n<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1677226\">\n<div id=\"fs-id1663685\">\n<h3>Graphing One Period of a Shifted Tangent Function<\/h3>\n<p id=\"fs-id1355894\">Graph one period of the function[latex]\\,y=-2\\mathrm{tan}\\left(\\pi x+\\pi \\right)\\,-1.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q826718\">Show Solution<\/span><\/p>\n<div id=\"q826718\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165133446963\">\n<li><em>Step 1.<\/em> The function is already written in the form[latex]\\,y=A\\mathrm{tan}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li><em>Step 2.<\/em>[latex]\\,A=-2,\\,[\/latex]so the stretching factor is[latex]\\,|A|=2.[\/latex]<\/li>\n<li><em>Step 3.<\/em>[latex]\\,B=\\pi ,\\,[\/latex]so the period is[latex]\\,P=\\frac{\\pi }{|B|}=\\frac{\\pi }{\\pi }=1.[\/latex]<\/li>\n<li><em>Step 4.<\/em>[latex]\\,C=-\\pi ,\\,[\/latex]so the phase shift is[latex]\\,\\frac{C}{B}=\\frac{-\\pi }{\\pi }=-1.[\/latex]<\/li>\n<li><em>Step 5-7.<\/em> The asymptotes are at[latex]\\,x=-\\frac{3}{2}\\,[\/latex]and[latex]\\,x=-\\frac{1}{2}\\,[\/latex]and the three recommended reference points are[latex]\\,\\left(-1.25,1\\right),\\,[\/latex][latex]\\left(-1,-1\\right),\\,[\/latex]and[latex]\\,\\left(-0.75,-3\\right).\\,[\/latex]The graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_005\">(Figure)<\/a>.\n<div id=\"Figure_06_02_005\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143622\/CNX_Precalc_Figure_06_02_005.jpg\" alt=\"A graph of one period of a shifted tangent function, with vertical asymptotes at x=-1.5 and x=-0.5.\" width=\"487\" height=\"193\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1700370\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1456051\">Note that this is a decreasing function because[latex]\\,A<0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1586707\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_03\">\n<div id=\"fs-id1379854\">\n<p id=\"fs-id1379856\">How would the graph in <a class=\"autogenerated-content\" href=\"#Example_06_02_02\">(Figure)<\/a> look different if we made[latex]\\,A=2\\,[\/latex]instead of[latex]\\,-2?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1394906\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1394906\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1394906\">It would be reflected across the line[latex]\\,y=-1,\\,[\/latex]becoming an increasing function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id672550\" class=\"precalculus howto\">\n<p id=\"fs-id1422588\"><strong>Given the graph of a tangent function, identify horizontal and vertical stretches.<\/strong><\/p>\n<ol id=\"fs-id1586539\" type=\"1\">\n<li>Find the period[latex]\\,P\\,[\/latex]from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\n<li>Write[latex]\\,f\\left(x\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{P}x\\right).[\/latex]<\/li>\n<li>Determine a convenient point[latex]\\,\\left(x,f\\left(x\\right)\\right)\\,[\/latex]on the given graph and use it to determine[latex]\\,A.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_03\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1430638\">\n<h3>Identifying the Graph of a Stretched Tangent<\/h3>\n<p id=\"fs-id1586016\">Find a formula for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_06_02_006\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_02_006\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143625\/CNX_Precalc_Figure_06_02_006.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.\" width=\"487\" height=\"256\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4. <\/strong>A stretched tangent function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1649014\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1649014\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1649014\">The graph has the shape of a tangent function.<\/p>\n<ul id=\"eip-id1165134188910\">\n<li><em>Step 1.<\/em> One cycle extends from \u20134 to 4, so the period is[latex]\\,P=8.\\,[\/latex]Since[latex]\\,P=\\frac{\\pi }{|B|},\\,[\/latex]we have[latex]\\,B=\\frac{\\pi }{P}=\\frac{\\pi }{8}.[\/latex]<\/li>\n<li><em>Step 2.<\/em> The equation must have the form[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{8}x\\right).[\/latex]<\/li>\n<li><em>Step 3.<\/em> To find the vertical stretch[latex]\\,A,[\/latex]we can use the point[latex]\\,\\left(2,2\\right).[\/latex]\n<div id=\"fs-id638950\" class=\"unnumbered aligncenter\">[latex]2=A\\mathrm{tan}\\left(\\frac{\\pi }{8}\\cdot 2\\right)=A\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p id=\"fs-id1536654\">Because[latex]\\,\\mathrm{tan}\\left(\\frac{\\pi }{4}\\right)=1,\\,[\/latex][latex]A=2.[\/latex]<\/p>\n<p id=\"fs-id1409295\">This function would have a formula[latex]\\,f\\left(x\\right)=2\\mathrm{tan}\\left(\\frac{\\pi }{8}x\\right).[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1655580\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_04\">\n<div id=\"fs-id1563795\">\n<p id=\"fs-id1563796\">Find a formula for the function in <a class=\"autogenerated-content\" href=\"#Figure_06_02_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_02_007\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143628\/CNX_Precalc_Figure_06_02_007.jpg\" alt=\"A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi\/4, -pi\/4, pi\/4, and 3pi\/4.\" width=\"487\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1662824\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1662824\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1662824\">[latex]g\\left(x\\right)=4\\mathrm{tan}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1653007\" class=\"bc-section section\">\n<h3>Analyzing the Graphs of <em>y<\/em> = sec <em>x<\/em> and <em>y<\/em> = csc<em>x<\/em><\/h3>\n<p id=\"fs-id1405395\">The <span class=\"no-emphasis\">secant<\/span> was defined by the <span class=\"no-emphasis\">reciprocal identity<\/span>[latex]\\,\\mathrm{sec}\\,x=\\frac{1}{\\mathrm{cos}\\,x}.\\,[\/latex]Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at[latex]\\,\\frac{\\pi }{2},\\,[\/latex][latex]\\frac{3\\pi }{2},\\,[\/latex]etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p id=\"fs-id1615608\">We can graph[latex]\\,y=\\mathrm{sec}\\,x\\,[\/latex]by observing the graph of the cosine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_008\">(Figure)<\/a>. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <span class=\"no-emphasis\">secant function<\/span> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.<\/p>\n<p id=\"fs-id1657904\">The secant graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.<\/p>\n<p id=\"fs-id1457737\">Note that, because cosine is an even function, secant is also an even function. That is,[latex]\\,\\mathrm{sec}\\left(-x\\right)=\\mathrm{sec}\\,x.[\/latex]<\/p>\n<div id=\"Figure_06_02_008\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143637\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" width=\"487\" height=\"379\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6. <\/strong>Graph of the secant function,[latex]\\,f\\left(x\\right)=\\mathrm{sec}x=\\frac{1}{\\mathrm{cos}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1584466\">As we did for the tangent function, we will again refer to the constant[latex]\\,|A|\\,[\/latex]as the stretching factor, not the amplitude.<\/p>\n<div id=\"fs-id1700128\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\n<ul id=\"fs-id1440349\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{2|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<li>The range is[latex]\\,\\left(-\\infty ,-|A|\\right]\\cup \\left[|A|,\\infty \\right).[\/latex]<\/li>\n<li>The vertical asymptotes occur at[latex]\\,x=\\frac{\\pi }{2|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\mathrm{sec}\\left(Bx\\right)\\,[\/latex]is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1380717\">Similar to the secant, the <span class=\"no-emphasis\">cosecant<\/span> is defined by the reciprocal identity[latex]\\,\\mathrm{csc}\\,x=\\frac{1}{\\mathrm{sin}\\,x}.\\,[\/latex]Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at[latex]\\,0,\\,[\/latex][latex]\\pi ,\\,[\/latex]etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p id=\"fs-id1413697\">We can graph[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]by observing the graph of the sine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">(Figure)<\/a>. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <span class=\"no-emphasis\">cosecant function<\/span> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\n<p id=\"fs-id1380921\">The cosecant graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\n<p id=\"fs-id1457402\">Note that, since sine is an odd function, the cosecant function is also an odd function. That is,[latex]\\,\\mathrm{csc}\\left(-x\\right)=\\mathrm{-csc}x.[\/latex]<\/p>\n<p id=\"eip-610\">The graph of cosecant, which is shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">(Figure)<\/a>, is similar to the graph of secant.<\/p>\n<div id=\"Figure_06_02_009\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143642\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" width=\"487\" height=\"377\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7. <\/strong>The graph of the cosecant function,[latex]\\,f\\left(x\\right)=\\mathrm{csc}x=\\frac{1}{\\mathrm{sin}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1683902\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>csc(<em>Bx<\/em>)<\/h3>\n<ul id=\"fs-id1425299\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\left(-\\infty ,-|A|\\right]\\cup \\left[|A|,\\infty \\right).[\/latex]<\/li>\n<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>[latex]y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1670966\" class=\"bc-section section\">\n<h3>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y<\/em>= csc <em>x<\/em><\/h3>\n<p id=\"fs-id1420053\">For shifted, compressed, and\/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the <span class=\"no-emphasis\">cosecant function<\/span> in the same way as for the secant and other functions.The equations become the following.<\/p>\n<div id=\"eip-341\">[latex]y=A\\mathrm{sec}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<div>[latex]y=A\\mathrm{csc}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<div id=\"eip-992\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\n<ul id=\"eip-id2701214\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<li>The range is[latex]\\,\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/li>\n<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"eip-439\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>csc(<em>Bx<\/em>\u2212<em>C<\/em>)+<em>D<\/em><\/h3>\n<ul id=\"eip-id1449110\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\,\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/li>\n<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,[\/latex]is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1420056\" class=\"precalculus howto\">\n<p id=\"fs-id1512502\"><strong>Given a function of the form[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"fs-id1690218\" type=\"1\">\n<li>Express the function given in the form[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\n<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Sketch the graph of[latex]\\,y=A\\mathrm{cos}\\left(Bx\\right).[\/latex]<\/li>\n<li>Use the reciprocal relationship between[latex]\\,y=\\mathrm{cos}\\,x\\,[\/latex]and[latex]\\,y=\\mathrm{sec}\\,x\\,[\/latex]to draw the graph of[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\n<li>Sketch the asymptotes.<\/li>\n<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_04\" class=\"textbox examples\">\n<div id=\"fs-id1698723\">\n<div id=\"fs-id1406755\">\n<h3>Graphing a Variation of the Secant Function<\/h3>\n<p id=\"fs-id1325018\">Graph one period of[latex]\\,f\\left(x\\right)=2.5\\mathrm{sec}\\left(0.4x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q331336\">Show Solution<\/span><\/p>\n<div id=\"q331336\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165135426343\">\n<li><em>Step 1.<\/em> The given function is already written in the general form,[latex]\\,y=A\\mathrm{sec}\\left(Bx\\right).[\/latex]<\/li>\n<li><em>Step 2.<\/em>[latex]\\,A=2.5\\,[\/latex]so the stretching factor is[latex]\\,\\text{2}\\text{.5}\\text{.}[\/latex]<\/li>\n<li><em>Step 3.<\/em>[latex]\\,B=0.4\\,[\/latex]so[latex]\\,P=\\frac{2\\pi }{0.4}=5\\pi .\\,[\/latex]The period is[latex]\\,5\\pi \\,[\/latex]units.<\/li>\n<li><em>Step 4.<\/em> Sketch the graph of the function[latex]\\,g\\left(x\\right)=2.5\\mathrm{cos}\\left(0.4x\\right).[\/latex]<\/li>\n<li><em>Step 5.<\/em> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.<\/li>\n<li><em>Steps 6\u20137.<\/em> Sketch two asymptotes at[latex]\\,x=1.25\\pi \\,[\/latex]and[latex]\\,x=3.75\\pi .\\,[\/latex]We can use two reference points, the local minimum at[latex]\\,\\left(0,2.5\\right)\\,[\/latex]and the local maximum at[latex]\\,\\left(2.5\\pi ,-2.5\\right).\\,[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_010\">(Figure)<\/a> shows the graph.\n<div id=\"Figure_06_02_010\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143645\/CNX_Precalc_Figure_06_02_010.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an upward facing prarbola and a downward facing parabola.\" width=\"487\" height=\"567\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1439628\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_05\">\n<div id=\"fs-id1690764\">\n<p id=\"fs-id1354617\">Graph one period of[latex]\\,f\\left(x\\right)=-2.5\\mathrm{sec}\\left(0.4x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1440114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1440114\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1440114\">This is a vertical reflection of the preceding graph because[latex]\\,A\\,[\/latex]is negative.<\/p>\n<p><span id=\"fs-id1380832\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143649\/CNX_Precalc_Figure_06_02_011.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1570004\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1440136\"><strong>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/strong><\/p>\n<p id=\"fs-id1420221\"><em>Yes. The range of<\/em>[latex]\\,f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]<em>is<\/em>[latex]\\left(-\\infty ,-|A|+D\\right]\\cup \\left[|A|+D,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1638418\" class=\"precalculus howto\">\n<p><strong>Given a function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"eip-id2515230\" type=\"1\">\n<li>Express the function given in the form[latex]\\,y=A\\,\\mathrm{sec}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\,\\mathrm{sec}\\left(Bx\\right)\\,.[\/latex]but shift it to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\n<li>Sketch the vertical asymptotes, which occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{2|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an odd integer.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1431671\">\n<div id=\"fs-id1431673\">\n<h3>Graphing a Variation of the Secant Function<\/h3>\n<p id=\"fs-id1537180\">Graph one period of[latex]\\,y=4\\mathrm{sec}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{2}\\right)+1.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q318489\">Show Solution<\/span><\/p>\n<div id=\"q318489\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"fs-id1906405\">\n<li><em>Step 1.<\/em> Express the function given in the form[latex]\\,y=4\\mathrm{sec}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{2}\\right)+1.[\/latex]<\/li>\n<li><em>Step 2.<\/em> The stretching\/compressing factor is[latex]|A|=4.[\/latex]<\/li>\n<li><em>Step 3.<\/em> The period is\n<div id=\"fs-id2071183\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{3}}\\hfill \\\\ \\text{ }=\\frac{2\\pi }{1}\\cdot \\frac{3}{\\pi }\\hfill \\\\ \\text{ }=6\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li><em>Step 4.<\/em> The phase shift is\n<div id=\"fs-id2341189\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{C}{B}=\\frac{\\frac{\\pi }{2}}{\\frac{\\pi }{3}}\\hfill \\\\ \\text{ }=\\frac{\\pi }{2}\\cdot \\frac{3}{\\pi }\\hfill \\\\ \\text{ }=1.5\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li><em>Step 5.<\/em> Draw the graph of [latex]\\,y=A\\mathrm{sec}\\left(Bx\\right),[\/latex]but shift it to the right by[latex]\\,\\frac{C}{B}=1.5\\,[\/latex]and up by[latex]\\,D=6.[\/latex]<\/li>\n<li><em>Step 6.<\/em> Sketch the vertical asymptotes, which occur at[latex]\\,x=0,x=3,[\/latex]and[latex]\\,x=6.\\,[\/latex]There is a local minimum at[latex]\\,\\left(1.5,5\\right)\\,[\/latex]and a local maximum at[latex]\\,\\left(4.5,-3\\right).\\,[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_012\">(Figure)<\/a> shows the graph.<\/li>\n<\/ul>\n<div id=\"Figure_06_02_012\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143653\/CNX_Precalc_Figure_06_02_012.jpg\" alt=\"\" width=\"487\" height=\"318\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1451968\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_06\">\n<div id=\"fs-id1346086\">\n<p id=\"fs-id1410617\">Graph one period of[latex]\\,f\\left(x\\right)=-6\\mathrm{sec}\\left(4x+2\\right)-8.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1439474\">Show Solution<\/span><\/p>\n<div id=\"q1439474\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1673799\"><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143655\/CNX_Precalc_Figure_06_02_013.jpg\" alt=\"A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi\/20 and one approximately at 3pi\/16.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1523838\" class=\"precalculus qa textbox shaded\">\n<p id=\"eip-id1165135487331\"><strong>The domain of[latex]\\,\\mathrm{csc}\\,x\\,[\/latex]was given to be all[latex]\\,x\\,[\/latex]such that[latex]\\,x\\ne k\\pi \\,[\/latex]for any integer[latex]\\,k.\\,[\/latex]<strong>Would the domain of<\/strong>[latex]\\,y=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,\\text{be}\\,x\\ne \\frac{C+k\\pi }{B}?[\/latex]<\/strong><\/p>\n<p id=\"fs-id1374943\"><em>Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function\u2019s input.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1428931\" class=\"precalculus howto\">\n<p id=\"fs-id1629286\"><strong>Given a function of the form[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right),\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"fs-id1395174\" type=\"1\">\n<li>Express the function given in the form[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\n<li>[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/li>\n<li>Use the reciprocal relationship between[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]and[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]to draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\n<li>Sketch the asymptotes.<\/li>\n<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1410488\">\n<div id=\"fs-id1410490\">\n<h3>Graphing a Variation of the Cosecant Function<\/h3>\n<p id=\"fs-id1425806\">Graph one period of[latex]\\,f\\left(x\\right)=-3\\mathrm{csc}\\left(4x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990021\">Show Solution<\/span><\/p>\n<div id=\"q990021\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165131907381\">\n<li><em>Step 1.<\/em> The given function is already written in the general form,[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right).[\/latex]<\/li>\n<li><em>Step 2.<\/em>[latex]\\,|A|=|-3|=3,[\/latex]so the stretching factor is 3.<\/li>\n<li><em>Step 3.<\/em>[latex]\\,B=4,[\/latex]so[latex]\\,P=\\frac{2\\pi }{4}=\\frac{\\pi }{2}.\\,[\/latex]The period is[latex]\\,\\frac{\\pi }{2}\\,[\/latex]units.<\/li>\n<li><em>Step 4.<\/em> Sketch the graph of the function[latex]\\,g\\left(x\\right)=-3\\mathrm{sin}\\left(4x\\right).[\/latex]<\/li>\n<li><em>Step 5.<\/em> Use the reciprocal relationship of the sine and cosecant functions to draw the <span class=\"no-emphasis\">cosecant function<\/span>.<\/li>\n<li><em>Steps 6\u20137.<\/em> Sketch three asymptotes at[latex]\\,x=0,\\,x=\\frac{\\pi }{4},\\,[\/latex]and[latex]\\,x=\\frac{\\pi }{2}.\\,[\/latex]We can use two reference points, the local maximum at[latex]\\,\\left(\\frac{\\pi }{8},-3\\right)\\,[\/latex]and the local minimum at[latex]\\,\\left(\\frac{3\\pi }{8},3\\right).[\/latex]<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_023\">(Figure)<\/a> shows the graph.\n<div id=\"Figure_06_02_023\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143703\/CNX_Precalc_Figure_06_02_023.jpg\" alt=\"A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi\/4, and x=pi\/2.\" width=\"487\" height=\"686\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1714936\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_07\">\n<div id=\"fs-id1422180\">\n<p id=\"fs-id1422181\">Graph one period of[latex]\\,f\\left(x\\right)=0.5\\mathrm{csc}\\left(2x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1150397\">Show Solution<\/span><\/p>\n<div id=\"q1150397\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1696881\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143715\/CNX_Precalc_Figure_06_02_023b.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id978295\" class=\"precalculus howto\">\n<p id=\"fs-id1046161\"><strong>Given a function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{csc}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"eip-id1453973\" type=\"1\">\n<li>Express the function given in the form[latex]\\,y=A\\mathrm{csc}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li>Identify the stretching\/compressing factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify[latex]\\,B\\,[\/latex]and determine the period,[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Identify[latex]\\,C\\,[\/latex]and determine the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by[latex]\\,D.[\/latex]<\/li>\n<li>Sketch the vertical asymptotes, which occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_07\" class=\"textbox examples\">\n<div id=\"fs-id1439464\">\n<div id=\"fs-id1439466\">\n<h3>Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\n<p id=\"fs-id1107734\">Sketch a graph of[latex]\\,y=2\\mathrm{csc}\\left(\\frac{\\pi }{2}x\\right)+1.\\,[\/latex]What are the domain and range of this function?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q61156\">Show Solution<\/span><\/p>\n<div id=\"q61156\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id3155416\">\n<li><em>Step 1.<\/em> Express the function given in the form[latex]\\,y=2\\mathrm{csc}\\left(\\frac{\\pi }{2}x\\right)+1.[\/latex]<\/li>\n<li><em>Step 2.<\/em> Identify the stretching\/compressing factor,[latex]\\,|A|=2.[\/latex]<\/li>\n<li><em>Step 3.<\/em> The period is[latex]\\,\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{2}}=\\frac{2\\pi }{1}\\cdot \\frac{2}{\\pi }=4.[\/latex]<\/li>\n<li><em>Step 4.<\/em> The phase shift is[latex]\\,\\frac{0}{\\frac{\\pi }{2}}=0.[\/latex]<\/li>\n<li><em>Step 5.<\/em> Draw the graph of[latex]\\,y=A\\mathrm{csc}\\left(Bx\\right)\\,[\/latex]but shift it up[latex]\\,D=1.[\/latex]<\/li>\n<li><em>Step 6.<\/em> Sketch the vertical asymptotes, which occur at[latex]\\,x=0,x=2,x=4.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id869117\">The graph for this function is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_014\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_02_014\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143731\/CNX_Precalc_Figure_06_02_014F.jpg\" alt=\"A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.\" width=\"487\" height=\"377\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11. <\/strong>A transformed cosecant function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1520970\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id728453\">The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of[latex]\\,f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)+1,[\/latex]shown as the orange dashed wave.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1588757\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_02_08\">\n<div id=\"fs-id1464306\">\n<p id=\"fs-id1464307\">Given the graph of[latex]\\,f\\left(x\\right)=2\\mathrm{cos}\\left(\\frac{\\pi }{2}x\\right)+1\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_015\">(Figure)<\/a>, sketch the graph of[latex]\\,g\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{2}x\\right)+1\\,[\/latex]on the same axes.<\/p>\n<div id=\"Figure_06_02_015\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143734\/CNX_Precalc_Figure_06_02_015.jpg\" alt=\"A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.\" width=\"488\" height=\"381\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1528623\">Show Solution<\/span><\/p>\n<div id=\"q1528623\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1662489\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143754\/CNX_Precalc_Figure_06_02_016.jpg\" alt=\"A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1232916\" class=\"bc-section section\">\n<h3>Analyzing the Graph of <em>y<\/em> = cot <em>x<\/em><\/h3>\n<p id=\"fs-id1585113\">The last trigonometric function we need to explore is <span class=\"no-emphasis\">cotangent<\/span>. The cotangent is defined by the <span class=\"no-emphasis\">reciprocal identity<\/span>[latex]\\,\\mathrm{cot}\\,x=\\frac{1}{\\mathrm{tan}\\,x}.\\,[\/latex]Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at[latex]\\,0,\\pi ,\\,[\/latex]etc. Since the output of the tangent function is all real numbers, the output of the <span class=\"no-emphasis\">cotangent function<\/span> is also all real numbers.<\/p>\n<p id=\"fs-id1333454\">We can graph[latex]\\,y=\\mathrm{cot}\\,x\\,[\/latex]by observing the graph of the tangent function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_017\">(Figure)<\/a>. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\n<p id=\"fs-id1430801\">The cotangent graph has vertical asymptotes at each value of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x=0;\\,[\/latex]we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,[latex]\\,\\mathrm{cot}\\,x\\,[\/latex]has vertical asymptotes at all values of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x=0,\\,[\/latex]and[latex]\\,\\mathrm{cot}\\,x=0\\,[\/latex]at all values of[latex]\\,x\\,[\/latex]where[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]has its vertical asymptotes.<\/p>\n<div id=\"Figure_06_02_017\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143757\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"487\" height=\"439\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13. <\/strong>The cotangent function<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1283383\">\n<h3>Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\n<ul id=\"fs-id1393165\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\,\\left(-\\infty ,\\infty \\right).[\/latex]<\/li>\n<li>The asymptotes occur at[latex]\\,x=\\frac{\\pi }{|B|}k,\\,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>[latex]y=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]is an odd function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1352247\" class=\"bc-section section\">\n<h3>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h3>\n<p id=\"fs-id1519587\">We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.<\/p>\n<div id=\"eip-1\">[latex]y=A\\mathrm{cot}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<div id=\"fs-id1658740\">\n<h3>Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\n<ul id=\"eip-id3983675\">\n<li>The stretching factor is[latex]\\,|A|.[\/latex]<\/li>\n<li>The period is[latex]\\,\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>The domain is[latex]\\,x\\ne \\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>The range is[latex]\\,\\left(\\mathrm{-\\infty },\\infty \\right).[\/latex]<\/li>\n<li>The vertical asymptotes occur at[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1526877\" class=\"precalculus howto\">\n<p id=\"fs-id1674147\"><strong>Given a modified cotangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right),[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"fs-id988921\" type=\"1\">\n<li>Express the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right).[\/latex]<\/li>\n<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right).[\/latex]<\/li>\n<li>Plot any two reference points.<\/li>\n<li>Use the reciprocal relationship between tangent and cotangent to draw the graph of[latex]\\,y=A\\mathrm{cot}\\left(Bx\\right).[\/latex]<\/li>\n<li>Sketch the asymptotes.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_08\" class=\"textbox examples\">\n<div id=\"fs-id918576\">\n<div id=\"fs-id918578\">\n<h3>Graphing Variations of the Cotangent Function<\/h3>\n<p id=\"fs-id1422599\">Determine the stretching factor, period, and phase shift of[latex]\\,y=3\\mathrm{cot}\\left(4x\\right),\\,[\/latex]and then sketch a graph.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q526668\">Show Solution<\/span><\/p>\n<div id=\"q526668\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165137777630\">\n<li><em>Step 1.<\/em> Expressing the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx\\right)\\,[\/latex]gives[latex]\\,f\\left(x\\right)=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/li>\n<li><em>Step 2.<\/em> The stretching factor is[latex]\\,|A|=3.[\/latex]<\/li>\n<li><em>Step 3.<\/em> The period is[latex]\\,P=\\frac{\\pi }{4}.[\/latex]<\/li>\n<li><em>Step 4.<\/em> Sketch the graph of[latex]\\,y=3\\mathrm{tan}\\left(4x\\right).[\/latex]<\/li>\n<li><em>Step 5.<\/em> Plot two reference points. Two such points are[latex]\\,\\left(\\frac{\\pi }{16},3\\right)\\,[\/latex]and[latex]\\,\\left(\\frac{3\\pi }{16},-3\\right).[\/latex]<\/li>\n<li><em>Step 6.<\/em> Use the reciprocal relationship to draw[latex]\\,y=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/li>\n<li><em>Step 7.<\/em> Sketch the asymptotes,[latex]\\,x=0,\\,\\,x=\\frac{\\pi }{4}.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1377143\">The orange graph in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_019\">(Figure)<\/a> shows[latex]\\,y=3\\mathrm{tan}\\left(4x\\right)\\,[\/latex]and the blue graph shows[latex]\\,y=3\\mathrm{cot}\\left(4x\\right).[\/latex]<\/p>\n<div id=\"Figure_06_02_019\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143801\/CNX_Precalc_Figure_06_02_019.jpg\" alt=\"A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi\/4 and pi\/4.\" width=\"487\" height=\"592\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1508070\" class=\"precalculus howto\">\n<p id=\"fs-id1692984\"><strong>Given a modified cotangent function of the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D,\\,[\/latex]graph one period.<\/strong><\/p>\n<ol id=\"fs-id1425287\" type=\"1\">\n<li>Express the function in the form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li>Identify the stretching factor,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify the period,[latex]\\,P=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>Identify the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,y=A\\mathrm{tan}\\left(Bx\\right)\\,[\/latex] shifted to the right by[latex]\\,\\frac{C}{B}\\,[\/latex]and up by[latex]\\,D.[\/latex]<\/li>\n<li>Sketch the asymptotes[latex]\\,x=\\frac{C}{B}+\\frac{\\pi }{|B|}k,[\/latex]where[latex]\\,k\\,[\/latex]is an integer.<\/li>\n<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_02_09\" class=\"textbox examples\">\n<div id=\"fs-id1700600\">\n<div id=\"fs-id1700602\">\n<h3>Graphing a Modified Cotangent<\/h3>\n<p id=\"fs-id1690330\">Sketch a graph of one period of the function[latex]\\,f\\left(x\\right)=4\\mathrm{cot}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q352430\">Show Solution<\/span><\/p>\n<div id=\"q352430\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165134038202\">\n<li><em>Step 1.<\/em> The function is already written in the general form[latex]\\,f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li><em>Step 2.<\/em>[latex]\\,A=4,[\/latex]so the stretching factor is 4.<\/li>\n<li><em>Step 3.<\/em>[latex]\\,B=\\frac{\\pi }{8},[\/latex]so the period is[latex]\\,P=\\frac{\\pi }{|B|}=\\frac{\\pi }{\\frac{\\pi }{8}}=8.[\/latex]<\/li>\n<li><em>Step 4.<\/em>[latex]\\,C=\\frac{\\pi }{2},[\/latex]so the phase shift is[latex]\\,\\frac{C}{B}=\\frac{\\frac{\\pi }{2}}{\\frac{\\pi }{8}}=4.[\/latex]<\/li>\n<li><em>Step 5.<\/em> We draw[latex]\\,f\\left(x\\right)=4\\mathrm{tan}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/li>\n<li><em>Step 6-7.<\/em> Three points we can use to guide the graph are[latex]\\,\\left(6,2\\right),\\left(8,-2\\right),\\,[\/latex]and[latex]\\,\\left(10,-6\\right).\\,[\/latex]We use the reciprocal relationship of tangent and cotangent to draw[latex]\\,f\\left(x\\right)=4\\mathrm{cot}\\left(\\frac{\\pi }{8}x-\\frac{\\pi }{2}\\right)-2.[\/latex]<\/li>\n<li><em>Step 8.<\/em> The vertical asymptotes are[latex]\\,x=4\\,[\/latex]and[latex]\\,x=12.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1445913\">The graph is shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2921&amp;action=edit#Figure_06_02_020\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_02_020\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143810\/CNX_Precalc_Figure_06_02_020.jpg\" alt=\"A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12.\" width=\"487\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 15. <\/strong>One period of a modified cotangent function<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1461208\" class=\"bc-section section\">\n<h3>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h3>\n<p id=\"fs-id1370674\">Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the <span class=\"no-emphasis\">tangent function<\/span>.<\/p>\n<div id=\"Example_06_02_10\" class=\"textbox examples\">\n<div id=\"fs-id1270016\">\n<div id=\"fs-id1270018\">\n<h3>Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\n<p id=\"fs-id1535059\">Suppose the function[latex]\\,y=5\\mathrm{tan}\\left(\\frac{\\pi }{4}t\\right)\\,[\/latex]marks the distance in the movement of a light beam from the top of a police car across a wall where[latex]\\,t\\,[\/latex]is the time in seconds and[latex]\\,y\\,[\/latex]is the distance in feet from a point on the wall directly across from the police car.<\/p>\n<ol id=\"fs-id1526727\" type=\"a\">\n<li>Find and interpret the stretching factor and period.<\/li>\n<li>Graph on the interval[latex]\\,\\left[0,5\\right].[\/latex]<\/li>\n<li>Evaluate[latex]\\,f\\left(1\\right)\\,[\/latex]and discuss the function\u2019s value at that input.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1658187\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1658187\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1658187\" type=\"a\">\n<li>We know from the general form of[latex]\\,y=A\\mathrm{tan}\\left(Bt\\right)\\,[\/latex]that[latex]\\,|A|\\,[\/latex]is the stretching factor and[latex]\\,\\frac{\\pi }{B}\\,[\/latex]is the period.\n<div id=\"Image_06_02_022\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143823\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 16.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id887356\">We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.<\/p>\n<p id=\"fs-id1651541\">The period is[latex]\\,\\frac{\\pi }{\\frac{\\pi }{4}}=\\frac{\\pi }{1}\\cdot \\frac{4}{\\pi }=4.\\,[\/latex]This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/p>\n<\/li>\n<li>To graph the function, we draw an asymptote at[latex]\\,t=2\\,[\/latex]and use the stretching factor and period. See <a class=\"autogenerated-content\" href=\"#Image_06_02_021\">(Figure)<\/a>\n<div id=\"Image_06_02_021\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143828\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 17.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>period:[latex]\\,f\\left(1\\right)=5\\mathrm{tan}\\left(\\frac{\\pi }{4}\\left(1\\right)\\right)=5\\left(1\\right)=5;\\,[\/latex]after 1 second, the beam of has moved 5 ft from the spot across from the police car.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1701460\" class=\"precalculus media\">\n<p id=\"fs-id1516989\">Access these online resources for additional instruction and practice with graphs of other trigonometric functions.<\/p>\n<ul id=\"fs-id1526866\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphtangent\">Graphing the Tangent<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphcscsec\">Graphing Cosecant and Secant<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphcot\">Graphing the Cotangent<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1526416\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133401597\" summary=\"..\">\n<tbody>\n<tr>\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\n<td>[latex]y=A\\,\\mathrm{tan}\\left(Bx-C\\right)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\n<td>[latex]y=A\\,\\mathrm{sec}\\left(Bx-C\\right)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cosecant function<\/td>\n<td>[latex]y=A\\,\\mathrm{csc}\\left(Bx-C\\right)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\n<td>[latex]y=A\\,\\mathrm{cot}\\left(Bx-C\\right)+D[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1693998\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1651799\">\n<li>The tangent function has period[latex]\\,\\pi .[\/latex]<\/li>\n<li>[latex]f\\left(x\\right)=A\\mathrm{tan}\\left(Bx-C\\right)+D\\,[\/latex]is a tangent with vertical and\/or horizontal stretch\/compression and shift. See <a class=\"autogenerated-content\" href=\"#Example_06_02_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_06_02_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_06_02_03\">(Figure)<\/a>.<\/li>\n<li>The secant and cosecant are both periodic functions with a period of[latex]\\,2\\pi .\\,[\/latex][latex]f\\left(x\\right)=A\\mathrm{sec}\\left(Bx-C\\right)+D\\,[\/latex]gives a shifted, compressed, and\/or stretched secant function graph. See <a class=\"autogenerated-content\" href=\"#Example_06_02_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_05\">(Figure)<\/a>.<\/li>\n<li>[latex]f\\left(x\\right)=A\\mathrm{csc}\\left(Bx-C\\right)+D\\,[\/latex]gives a shifted, compressed, and\/or stretched cosecant function graph. See <a class=\"autogenerated-content\" href=\"#Example_06_02_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_07\">(Figure)<\/a>.<\/li>\n<li>The cotangent function has period[latex]\\,\\pi \\,[\/latex]and vertical asymptotes at[latex]\\,0,\u00b1\\pi ,\u00b12\\pi ,....[\/latex]<\/li>\n<li>The range of cotangent is[latex]\\,\\left(-\\infty ,\\infty \\right),\\,[\/latex]and the function is decreasing at each point in its range.<\/li>\n<li>The cotangent is zero at[latex]\\,\u00b1\\frac{\\pi }{2},\u00b1\\frac{3\\pi }{2},....[\/latex]<\/li>\n<li>[latex]f\\left(x\\right)=A\\mathrm{cot}\\left(Bx-C\\right)+D\\,[\/latex]is a cotangent with vertical and\/or horizontal stretch\/compression and shift. See <a class=\"autogenerated-content\" href=\"#Example_06_02_08\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_02_09\">(Figure)<\/a>.<\/li>\n<li>Real-world scenarios can be solved using graphs of trigonometric functions. See <a class=\"autogenerated-content\" href=\"#Example_06_02_10\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id878869\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id878873\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1407115\">\n<div id=\"fs-id1407118\">\n<p id=\"fs-id1407119\">Explain how the graph of the sine function can be used to graph[latex]\\,y=\\mathrm{csc}\\,x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1439607\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1439607\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1439607\">Since[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]is the reciprocal function of[latex]\\,y=\\mathrm{sin}\\,x,\\,[\/latex]you can plot the reciprocal of the coordinates on the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]to obtain the <em>y<\/em>-coordinates of[latex]\\,y=\\mathrm{csc}\\,x.\\,[\/latex]The <em>x<\/em>-intercepts of the graph[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]are the vertical asymptotes for the graph of[latex]\\,y=\\mathrm{csc}\\,x.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1513360\">\n<div id=\"fs-id1513362\">\n<p id=\"fs-id1454682\">How can the graph of[latex]\\,y=\\mathrm{cos}\\,x\\,[\/latex]be used to construct the graph of[latex]\\,y=\\mathrm{sec}\\,x?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1421392\">\n<div id=\"fs-id1421394\">\n<p id=\"fs-id1421396\">Explain why the period of[latex]\\,\\mathrm{tan}\\,x\\,[\/latex]is equal to[latex]\\,\\pi .[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1461033\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1461033\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1461033\">Answers will vary. Using the unit circle, one can show that[latex]\\,\\mathrm{tan}\\left(x+\\pi \\right)=\\mathrm{tan}\\,x.\\,[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1409386\">\n<div id=\"fs-id1409388\">\n<p id=\"fs-id1351021\">Why are there no intercepts on the graph of[latex]\\,y=\\mathrm{csc}\\,x?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1529165\">\n<div id=\"fs-id1529167\">\n<p id=\"fs-id1529169\">How does the period of[latex]\\,y=\\mathrm{csc}\\,x\\,[\/latex]compare with the period of[latex]\\,y=\\mathrm{sin}\\,x?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1365332\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1365332\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1365332\">The period is the same:[latex]\\,2\\pi .[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1608979\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1608984\">For the following exercises, match each trigonometric function with one of the following graphs.<\/p>\n<p><span id=\"Figure_06_02_201a\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143830\/CNX_Precalc_Figure_06_02_201a.jpg\" alt=\"Trigonometric graph of tangent of x.\" \/><\/span><span id=\"Figure_06_02_201b\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143832\/CNX_Precalc_Figure_06_02_201b.jpg\" alt=\"Trigonometric graph of secant of x.\" \/><\/span><span id=\"Figure_06_02_201c\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143837\/CNX_Precalc_Figure_06_02_201c.jpg\" alt=\"Trigonometric graph of cosecant of x.\" \/><\/span><span id=\"Figure_06_02_201d\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143844\/CNX_Precalc_Figure_06_02_201d.jpg\" alt=\"Trigonometric graph of cotangent of x.\" \/><\/span><\/p>\n<div id=\"fs-id1600438\">\n<div id=\"fs-id1600440\">\n<p id=\"fs-id1600443\">[latex]f\\left(x\\right)=\\mathrm{tan}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1402861\">\n<div id=\"fs-id1402863\">\n<p id=\"fs-id1402865\">[latex]f\\left(x\\right)=\\mathrm{sec}\\,x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1512360\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1512360\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1512360\">IV<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1512365\">\n<div id=\"fs-id1451983\">\n<p id=\"fs-id1451985\">[latex]f\\left(x\\right)=\\mathrm{csc}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1434924\">\n<div id=\"fs-id1434926\">\n<p id=\"fs-id1673834\">[latex]f\\left(x\\right)=\\mathrm{cot}\\,x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1410903\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1410903\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1410903\">III<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1365640\">For the following exercises, find the period and horizontal shift of each of the functions.<\/p>\n<div id=\"fs-id1365643\">\n<div id=\"fs-id1365645\">\n<p id=\"fs-id1562031\">[latex]f\\left(x\\right)=2\\mathrm{tan}\\left(4x-32\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1662387\">\n<div id=\"fs-id1662389\">\n<p id=\"fs-id1671047\">[latex]h\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{4}\\left(x+1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1581685\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1581685\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1581685\">period: 8; horizontal shift: 1 unit to left<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1405674\">\n<div id=\"fs-id1405676\">\n<p id=\"fs-id1405678\">[latex]m\\left(x\\right)=6\\mathrm{csc}\\left(\\frac{\\pi }{3}x+\\pi \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1652255\">\n<div id=\"fs-id1652257\">\n<p id=\"fs-id1652259\">If[latex]\\,\\mathrm{tan}\\,x=-1.5,\\,[\/latex]find[latex]\\,\\mathrm{tan}\\left(-x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1429264\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1429264\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1429264\">1.5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1422443\">\n<div id=\"fs-id1422445\">\n<p id=\"fs-id1422447\">If[latex]\\,\\mathrm{sec}\\,x=2,\\,[\/latex]find[latex]\\,\\mathrm{sec}\\left(-x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1356505\">\n<div id=\"fs-id1456034\">\n<p id=\"fs-id1456037\">If[latex]\\,\\mathrm{csc}\\,x=-5,\\,[\/latex]find[latex]\\,\\mathrm{csc}\\left(-x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1375607\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1375607\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1375607\">5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1404272\">\n<div id=\"fs-id1404274\">\n<p id=\"fs-id1428797\">If[latex]\\,x\\mathrm{sin}\\,x=2,\\,[\/latex]find[latex]\\,\\left(-x\\right)\\mathrm{sin}\\left(-x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1508271\">For the following exercises, rewrite each expression such that the argument[latex]\\,x\\,[\/latex]is positive.<\/p>\n<div id=\"fs-id1351072\">\n<div id=\"fs-id1351074\">\n<p id=\"fs-id1351076\">[latex]\\mathrm{cot}\\left(-x\\right)\\mathrm{cos}\\left(-x\\right)+\\mathrm{sin}\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1407073\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1407073\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1407073\">[latex]-\\mathrm{cot}x\\mathrm{cos}x-\\mathrm{sin}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1458413\">\n<div id=\"fs-id1458415\">\n<p id=\"fs-id1458417\">[latex]\\mathrm{cos}\\left(-x\\right)+\\mathrm{tan}\\left(-x\\right)\\mathrm{sin}\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1513346\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1283129\">For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.<\/p>\n<div id=\"fs-id1283134\">\n<div id=\"fs-id1434575\">\n<p id=\"fs-id1434577\">[latex]f\\left(x\\right)=2\\mathrm{tan}\\left(4x-32\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q46974\">Show Solution<\/span><\/p>\n<div id=\"q46974\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1644075\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143846\/CNX_Precalc_Figure_06_02_202.jpg\" alt=\"A graph of two periods of a modified tangent function. There are two vertical asymptotes.\" \/><\/span><\/p>\n<p id=\"fs-id1656875\">stretching factor: 2; period:[latex]\\text{ }\\frac{\\pi }{4};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{1}{4}\\left(\\frac{\\pi }{2}+\\pi k\\right)+8,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1656875\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1422235\">\n<p id=\"fs-id1422238\">[latex]\\,h\\left(x\\right)=2\\mathrm{sec}\\left(\\frac{\\pi }{4}\\left(x+1\\right)\\right)\\,[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1435841\">\n<div id=\"fs-id1419768\">\n<p id=\"fs-id1419770\">[latex]m\\left(x\\right)=6\\mathrm{csc}\\left(\\frac{\\pi }{3}x+\\pi \\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278253\">Show Solution<\/span><\/p>\n<div id=\"q278253\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1375745\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143851\/CNX_Precalc_Figure_06_02_204.jpg\" alt=\"A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.\" \/><\/span><\/p>\n<p id=\"fs-id1512258\">stretching factor: 6; period: 6; asymptotes:[latex]\\text{ }x=3k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1512258\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1562828\">\n<div id=\"fs-id1562830\">\n<p id=\"fs-id1562832\">[latex]j\\left(x\\right)=\\mathrm{tan}\\left(\\frac{\\pi }{2}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1581814\">\n<div id=\"fs-id1581816\">\n<p id=\"fs-id1581818\">[latex]p\\left(x\\right)=\\mathrm{tan}\\left(x-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q528553\">Show Solution<\/span><\/p>\n<div id=\"q528553\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1365984\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143854\/CNX_Precalc_Figure_06_02_206.jpg\" alt=\"A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.\" \/><\/span><\/p>\n<p id=\"fs-id1419808\">stretching factor: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1419808\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1385170\">\n<div id=\"fs-id1385172\">\n<p id=\"fs-id1385174\">[latex]f\\left(x\\right)=4\\mathrm{tan}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1506812\">\n<div id=\"fs-id1506814\">\n<p id=\"fs-id1506816\">[latex]f\\left(x\\right)=\\mathrm{tan}\\left(x+\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q156444\">Show Solution<\/span><\/p>\n<div id=\"q156444\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1354983\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143857\/CNX_Precalc_Figure_06_02_208.jpg\" alt=\"A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.\" \/><\/span><\/p>\n<p id=\"fs-id1528080\">Stretching factor: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1528080\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1364876\">\n<div id=\"fs-id1364878\">\n<p id=\"fs-id1364880\">[latex]f\\left(x\\right)=\\pi \\mathrm{tan}\\left(\\pi x-\\pi \\right)-\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1506136\">\n<div id=\"fs-id1655270\">\n<p id=\"fs-id1655272\">[latex]f\\left(x\\right)=2\\mathrm{csc}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q970468\">Show Solution<\/span><\/p>\n<div id=\"q970468\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1455973\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143903\/CNX_Precalc_Figure_06_02_210.jpg\" alt=\"A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.\" \/><\/span><\/p>\n<p id=\"fs-id1489928\">stretching factor: 2; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1489928\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1402973\">\n<div id=\"fs-id1402975\">\n<p id=\"fs-id1405719\">[latex]f\\left(x\\right)=-\\frac{1}{4}\\mathrm{csc}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1699640\">\n<div id=\"fs-id1408399\">\n<p id=\"fs-id1408401\">[latex]f\\left(x\\right)=4\\mathrm{sec}\\left(3x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q999274\">Show Solution<\/span><\/p>\n<div id=\"q999274\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1704371\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143909\/CNX_Precalc_Figure_06_02_212.jpg\" alt=\"A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi\/2, -pi\/6, pi\/6, and pi\/2.\" \/><\/span><\/p>\n<p id=\"fs-id1583862\">stretching factor: 4; period:[latex]\\text{ }\\frac{2\\pi }{3};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{6}k,\\text{ where }k\\text{ is an odd integer}[\/latex]<\/p>\n<p id=\"fs-id1583862\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1705843\">\n<div id=\"fs-id1705846\">\n<p id=\"fs-id1705848\">[latex]f\\left(x\\right)=-3\\mathrm{cot}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1375944\">\n<div id=\"fs-id1420097\">\n<p id=\"fs-id1420099\">[latex]f\\left(x\\right)=7\\mathrm{sec}\\left(5x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q643578\">Show Solution<\/span><\/p>\n<div id=\"q643578\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1429978\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143916\/CNX_Precalc_Figure_06_02_214.jpg\" alt=\"A graph of two periods of a modified secant function. There are four vertical asymptotes all pi\/5 apart.\" \/><\/span><\/p>\n<p id=\"fs-id1582375\">stretching factor: 7; period:[latex]\\text{ }\\frac{2\\pi }{5};\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{10}k,\\text{ where }k\\text{ is an odd integer}[\/latex]<\/p>\n<p id=\"fs-id1582375\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1655075\">\n<div id=\"fs-id1655077\">\n<p id=\"fs-id1417308\">[latex]f\\left(x\\right)=\\frac{9}{10}\\mathrm{csc}\\left(\\pi x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1523811\">\n<div id=\"fs-id1523813\">\n<p id=\"fs-id1523815\">[latex]f\\left(x\\right)=2\\mathrm{csc}\\left(x+\\frac{\\pi }{4}\\right)-1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125458\">Show Solution<\/span><\/p>\n<div id=\"q125458\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1404603\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143918\/CNX_Precalc_Figure_06_02_216.jpg\" alt=\"A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.\" \/><\/span><\/p>\n<p id=\"fs-id1361638\">stretching factor: 2; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=-\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1361638\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1434375\">\n<div id=\"fs-id1434377\">\n<p id=\"fs-id1434379\">[latex]f\\left(x\\right)=-\\mathrm{sec}\\left(x-\\frac{\\pi }{3}\\right)-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1614813\">\n<div id=\"fs-id1614815\">\n<p id=\"fs-id1614818\">[latex]f\\left(x\\right)=\\frac{7}{5}\\mathrm{csc}\\left(x-\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q538264\">Show Solution<\/span><\/p>\n<div id=\"q538264\" class=\"hidden-answer\" style=\"display: none\">\n<p><span id=\"fs-id1446559\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143921\/CNX_Precalc_Figure_06_02_218.jpg\" alt=\"A graph of a modified cosecant function. Four vertical asymptotes.\" \/><\/span><\/p>\n<p id=\"fs-id1375388\">stretching factor:[latex]\\text{ }\\frac{7}{5};\\text{ }[\/latex]period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]asymptotes:[latex]\\text{ }x=\\frac{\\pi }{4}+\\pi k,\\text{ where }k\\text{ is an integer}[\/latex]<\/p>\n<p id=\"fs-id1375388\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1409971\">\n<div id=\"fs-id1409974\">\n<p id=\"fs-id1409976\">[latex]f\\left(x\\right)=5\\left(\\mathrm{cot}\\left(x+\\frac{\\pi }{2}\\right)-3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1464121\">For the following exercises, find and graph two periods of the periodic function with the given stretching factor,[latex]\\,|A|,\\,[\/latex]period, and phase shift.<\/p>\n<div>\n<div>\n<p id=\"fs-id1649334\">A tangent curve,[latex]\\,A=1,\\,[\/latex]period of[latex]\\,\\frac{\\pi }{3};\\,[\/latex]and phase shift[latex]\\,\\left(h,\\,k\\right)=\\left(\\frac{\\pi }{4},2\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1694857\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1694857\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1694857\">[latex]y=\\mathrm{tan}\\left(3\\left(x-\\frac{\\pi }{4}\\right)\\right)+2[\/latex]<\/p>\n<p><span id=\"fs-id1649707\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143924\/CNX_Precalc_Figure_06_02_220.jpg\" alt=\"A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi\/4 and pi\/12.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1658055\">\n<div id=\"fs-id1440828\">\n<p id=\"fs-id1440830\">A tangent curve,[latex]\\,A=-2,\\,[\/latex]period of[latex]\\,\\frac{\\pi }{4},\\,[\/latex]and phase shift[latex]\\,\\left(h,\\,k\\right)=\\left(-\\frac{\\pi }{4},\\,-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1588386\">For the following exercises, find an equation for the graph of each function.<\/p>\n<div id=\"fs-id1588389\">\n<div id=\"fs-id1588390\"><span id=\"fs-id1513948\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143931\/CNX_Precalc_Figure_06_02_222.jpg\" alt=\"A graph of two periods of a modified cosecant function, with asymptotes at multiples of pi\/2.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1706414\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1706414\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1706414\">[latex]f\\left(x\\right)=\\mathrm{csc}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1464236\">\n<div id=\"fs-id1464237\"><span id=\"fs-id1537949\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143933\/CNX_Precalc_Figure_06_02_223.jpg\" alt=\"A graph of a modified cotangent function. Vertical asymptotes at x=-1 and x=0 and x=1.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1409010\">\n<div id=\"fs-id1409012\"><span id=\"fs-id1409017\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143936\/CNX_Precalc_Figure_06_02_224.jpg\" alt=\"A graph of a modified cosecant function. Vertical asymptotes at multiples of pi\/4.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1405484\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1405484\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1405484\">[latex]f\\left(x\\right)=\\mathrm{csc}\\left(4x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1465129\">\n<div id=\"fs-id1465130\"><span id=\"fs-id1465137\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143949\/CNX_Precalc_Figure_06_02_225.jpg\" alt=\"A graph of a modified tangent function. Vertical asymptotes at -pi\/8 and 3pi\/8.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1653766\">\n<div id=\"fs-id1653767\"><span id=\"fs-id1445558\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143954\/CNX_Precalc_Figure_06_02_226.jpg\" alt=\"A graph of a modified cosecant function. Vertical asymptotyes at multiples of pi.\" \/><\/span><\/div>\n<div id=\"fs-id705620\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id705620\">Show Solution<\/span><\/p>\n<div id=\"qfs-id705620\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id705621\">[latex]f\\left(x\\right)=2\\mathrm{csc}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1530295\">\n<div id=\"fs-id1530296\"><span id=\"fs-id1695827\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143957\/CNX_Precalc_Figure_06_02_227.jpg\" alt=\"A graph of a modified secant function. Four vertical asymptotes.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1664599\">\n<div id=\"fs-id1664600\"><span id=\"fs-id1664606\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143959\/CNX_Precalc_Figure_06_02_228.jpg\" alt=\"graph of two periods of a modified tangent function. Vertical asymptotes at x=-0.005 and x=0.005.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1460872\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1460872\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1460872\">[latex]f\\left(x\\right)=\\frac{1}{2}\\mathrm{tan}\\left(100\\pi x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1678301\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1678306\">For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input[latex]\\,\\mathrm{csc}\\,x\\,[\/latex]as[latex]\\,\\frac{1}{\\mathrm{sin}\\,x}.[\/latex]<\/p>\n<div id=\"fs-id1691184\">\n<div id=\"fs-id1691186\">\n<p id=\"fs-id1691188\">[latex]f\\left(x\\right)=|\\mathrm{csc}\\left(x\\right)|[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1690901\">\n<div id=\"fs-id1447025\">\n<p id=\"fs-id1447027\">[latex]f\\left(x\\right)=|\\mathrm{cot}\\left(x\\right)|[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1656302\">Show Solution<\/span><\/p>\n<div id=\"q1656302\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1656308\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144004\/CNX_Precalc_Figure_06_02_230.jpg\" alt=\"A graph of the absolute value of the cotangent function. Range is 0 to infinity.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1699978\">\n<div id=\"fs-id1699980\">\n<p id=\"fs-id1699982\">[latex]f\\left(x\\right)={2}^{\\mathrm{csc}\\left(x\\right)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1451630\">[latex]f\\left(x\\right)=\\frac{\\mathrm{csc}\\left(x\\right)}{\\mathrm{sec}\\left(x\\right)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1420323\">Show Solution<\/span><\/p>\n<div id=\"q1420323\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1420330\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144011\/CNX_Precalc_Figure_06_02_232.jpg\" alt=\"A graph of tangent of x.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1521332\">\n<div id=\"fs-id1586756\">\n<p id=\"fs-id1586758\">Graph[latex]\\,f\\left(x\\right)=1+{\\mathrm{sec}}^{2}\\left(x\\right)-{\\mathrm{tan}}^{2}\\left(x\\right).\\,[\/latex]What is the function shown in the graph?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1600902\">\n<div id=\"fs-id1600904\">\n<p id=\"fs-id1600906\">[latex]f\\left(x\\right)=\\mathrm{sec}\\left(0.001x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1586275\">Show Solution<\/span><\/p>\n<div id=\"q1586275\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1586282\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144019\/CNX_Precalc_Figure_06_02_234.jpg\" alt=\"A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1432442\">\n<div id=\"fs-id1432444\">\n<p id=\"fs-id1671603\">[latex]f\\left(x\\right)=\\mathrm{cot}\\left(100\\pi x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1536091\">\n<div id=\"fs-id1536094\">\n<p id=\"fs-id1536096\">[latex]f\\left(x\\right)={\\mathrm{sin}}^{2}x+{\\mathrm{cos}}^{2}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1631004\">Show Solution<\/span><\/p>\n<div id=\"q1631004\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1631010\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144024\/CNX_Precalc_Figure_06_02_241.jpg\" alt=\"A graph of y=1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1526988\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1526993\">\n<div id=\"fs-id1526995\">\n<p id=\"fs-id1405403\">The function[latex]\\,f\\left(x\\right)=20\\mathrm{tan}\\left(\\frac{\\pi }{10}x\\right)\\,[\/latex]marks the distance in the movement of a light beam from a police car across a wall for time[latex]\\,x,\\,[\/latex]in seconds, and distance[latex]\\,f\\left(x\\right),[\/latex]<br \/>\nin feet.<\/p>\n<ol id=\"fs-id1384936\" type=\"a\">\n<li>Graph on the interval[latex]\\,\\left[0,\\,5\\right].[\/latex]<\/li>\n<li>Find and interpret the stretching factor, period, and asymptote.<\/li>\n<li>Evaluate[latex]\\,f\\left(1\\right)\\,[\/latex]and[latex]\\,f\\left(2.5\\right)\\,[\/latex]and discuss the function\u2019s values at those inputs.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1513694\">\n<div id=\"fs-id1513696\">\n<p id=\"fs-id1513698\">Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let[latex]\\,x,\\,[\/latex]measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and[latex]\\,x\\,[\/latex]is measured negative to the left and positive to the right. (See <a class=\"autogenerated-content\" href=\"#Figure_06_02_237\">(Figure)<\/a>.) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance[latex]\\,d\\left(x\\right),\\,[\/latex]in kilometers, from the fisherman to the boat is given by the function[latex]\\,d\\left(x\\right)=1.5\\mathrm{sec}\\left(x\\right).[\/latex]<\/p>\n<ol id=\"fs-id1434023\" type=\"a\">\n<li>What is a reasonable domain for[latex]\\,d\\left(x\\right)?[\/latex]<\/li>\n<li>Graph[latex]\\,d\\left(x\\right)\\,[\/latex]on this domain.<\/li>\n<li>Find and discuss the meaning of any vertical asymptotes on the graph of[latex]\\,d\\left(x\\right).[\/latex]<\/li>\n<li>Calculate and interpret[latex]\\,d\\left(-\\frac{\\pi }{3}\\right).\\,[\/latex]Round to the second decimal place.<\/li>\n<li>Calculate and interpret[latex]\\,d\\left(\\frac{\\pi }{6}\\right).\\,[\/latex]Round to the second decimal place.<\/li>\n<li>What is the minimum distance between the fisherman and the boat? When does this occur?<\/li>\n<\/ol>\n<div id=\"Figure_06_02_237\" class=\"medium\">\n<div style=\"width: 475px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144035\/CNX_Precalc_Figure_06_02_237.jpg\" alt=\"An illustration of a man and the distance he is away from a boat.\" width=\"465\" height=\"280\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 18.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1700717\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1700717\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1700717\" type=\"a\">\n<li>[latex]\\,\\left(-\\frac{\\pi }{2},\\,\\frac{\\pi }{2}\\right);\\,[\/latex]<\/li>\n<li><span id=\"fs-id1532106\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144038\/CNX_Precalc_Figure_06_02_238.jpg\" alt=\"A graph of a half period of a secant function. Vertical asymptotes at x=-pi\/2 and pi\/2.\" \/><\/span><\/li>\n<li>[latex]\\,x=-\\frac{\\pi }{2}\\,[\/latex]and[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]the distance grows without bound as[latex]\\,|x|[\/latex]approaches[latex]\\,\\frac{\\pi }{2}\\,[\/latex]\u2014i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;<\/li>\n<li>3; when[latex]\\,x=-\\frac{\\pi }{3},\\,[\/latex]the boat is 3 km away;<\/li>\n<li>1.73; when[latex]\\,x=\\frac{\\pi }{6},\\,[\/latex]the boat is about 1.73 km away;<\/li>\n<li>1.5 km; when[latex]\\,x=0\\,[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1431336\">\n<div id=\"fs-id1431338\">\n<p id=\"fs-id1431340\">A laser rangefinder is locked on a comet approaching Earth. The distance[latex]\\,g\\left(x\\right),\\,[\/latex]in kilometers, of the comet after[latex]\\,x\\,[\/latex]days, for[latex]\\,x\\,[\/latex]in the interval 0 to 30 days, is given by[latex]\\,g\\left(x\\right)=250,000\\mathrm{csc}\\left(\\frac{\\pi }{30}x\\right).[\/latex]<\/p>\n<ol id=\"fs-id1526530\" type=\"a\">\n<li>Graph[latex]\\,g\\left(x\\right)\\,[\/latex]on the interval[latex]\\,\\left[0,\\,35\\right].[\/latex]<\/li>\n<li>Evaluate[latex]\\,g\\left(5\\right)\\,[\/latex]<br \/>\nand interpret the information.<\/li>\n<li>What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?<\/li>\n<li>Find and discuss the meaning of any vertical asymptotes.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1601867\">\n<div id=\"fs-id1601869\">\n<p id=\"fs-id1422576\">A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after[latex]\\,x\\,[\/latex]seconds is[latex]\\,\\frac{\\pi }{120}x.[\/latex]<\/p>\n<ol id=\"fs-id1614554\" type=\"a\">\n<li>Write a function expressing the altitude[latex]\\,h\\left(x\\right),\\,[\/latex]in miles, of the rocket above the ground after[latex]\\,x\\,[\/latex]seconds. Ignore the curvature of the Earth.<\/li>\n<li>Graph[latex]\\,h\\left(x\\right)\\,[\/latex]on the interval[latex]\\,\\left(0,\\,60\\right).[\/latex]<\/li>\n<li>Evaluate and interpret the values[latex]\\,h\\left(0\\right)\\,[\/latex]and[latex]\\,h\\left(30\\right).[\/latex]<\/li>\n<li>What happens to the values of[latex]\\,h\\left(x\\right)\\,[\/latex]as [latex]\\,x\\,[\/latex]<br \/>\napproaches 60 seconds? Interpret the meaning of this in terms of the problem.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1523906\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1523906\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1523906\" type=\"a\">\n<li>[latex]h\\left(x\\right)=2\\mathrm{tan}\\left(\\frac{\\pi }{120}x\\right);[\/latex]<\/li>\n<li><span id=\"fs-id1601972\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144041\/CNX_Precalc_Figure_06_02_240.jpg\" alt=\"An exponentially increasing function with a vertical asymptote at x=60.\" \/><\/span><\/li>\n<li>[latex]h\\left(0\\right)=0:\\,[\/latex]after 0 seconds, the rocket is 0 mi above the ground;[latex]h\\left(30\\right)=2:\\,[\/latex]after 30 seconds, the rockets is 2 mi high;<\/li>\n<li>As[latex]\\,x\\,[\/latex]approaches 60 seconds, the values of[latex]\\,h\\left(x\\right)\\,[\/latex]grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2921\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Algebra and Trigonometry. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\">http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":53384,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Algebra and Trigonometry\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"OpenStax CNX\",\"url\":\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2921","chapter","type-chapter","status-publish","hentry"],"part":2784,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2921","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/users\/53384"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2921\/revisions"}],"predecessor-version":[{"id":3685,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2921\/revisions\/3685"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/parts\/2784"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2921\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/media?parent=2921"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapter-type?post=2921"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/contributor?post=2921"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/license?post=2921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}