{"id":1377,"date":"2023-06-05T14:51:07","date_gmt":"2023-06-05T14:51:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/graphs-of-the-other-trigonometric-functions\/"},"modified":"2023-06-05T14:51:07","modified_gmt":"2023-06-05T14:51:07","slug":"graphs-of-the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/graphs-of-the-other-trigonometric-functions\/","title":{"raw":"Graphs of the Other Trigonometric Functions","rendered":"Graphs of the Other Trigonometric Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li style=\"font-weight: 400;\">Analyze the graph of \u2009y=tan\u2009x and y=cot x.<\/li>\n \t<li style=\"font-weight: 400;\">Graph variations of \u2009y=tan\u2009x and y=cot x.<\/li>\n \t<li>Determine a function formula from a tangent or cotangent graph.<\/li>\n \t<li style=\"font-weight: 400;\">Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n \t<li style=\"font-weight: 400;\">Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n \t<li>Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\nWe will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that\n<div>\n<div style=\"text-align: center;\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\n<\/div>\nThe <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\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.\n\nWe can determine whether tangent is an odd or even function by using the definition of tangent.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan(\u2212x)&amp;=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} &amp;&amp; \\text{Definition of tangent.} \\\\ &amp;=\\frac{\u2212\\sin x}{\\cos x} &amp;&amp; \\text{Sine is an odd function, cosine is even.} \\\\ &amp;=\u2212\\frac{\\sin x}{\\cos x} &amp;&amp; \\text{The quotient of an odd and an even function is odd.} \\\\ &amp;=\u2212\\tan x &amp;&amp; \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\nTherefore, 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 the table below.\n<table id=\"Table_06_02_00\" style=\"width: 1035px;\" 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 style=\"width: 30px;\"><em><strong> x <\/strong><\/em><\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30px;\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">\u20131<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">1<\/td>\n<td style=\"width: 80px;\">[latex]\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThese 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 x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.\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><em><strong> x <\/strong><\/em><\/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> tan <em>x <\/em><\/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>\nAs <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.\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><em><strong> x <\/strong><\/em><\/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> tan <em>x <\/em><\/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>\nWe can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. Figure 1&nbsp;represents the graph of [latex]y=\\tan x[\/latex]. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163804\/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\"> <b>Figure 1.<\/b> Graph of the tangent function[\/caption]<\/figure>\n<h2>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h2>\nAs with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\n<\/div>\nWe can identify horizontal and vertical stretches and compressions using values of A and B. 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.\n\nBecause 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 A.\n<div class=\"textbox\"><header>\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\n<\/header>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>| .<\/li>\n \t<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> is an integer.<\/li>\n \t<li>The range is [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/li>\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\nWe can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> 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](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\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\n<div>\n<div style=\"text-align: center;\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\n<\/div>\nbecause &nbsp;[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Identify the stretching factor, |A|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>Draw vertical asymptotes at &nbsp;[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\n \t<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\n \t<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]&nbsp;(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Sketching a Compressed Tangent<\/h3>\nSketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].\n\n[reveal-answer q=\"302986\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"302986\"]\n\nFirst, we identify <em>A<\/em> and B.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163807\/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.\" width=\"487\" height=\"113\"> <b>Figure 2<\/b>[\/caption]\n\nBecause [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have\n<p style=\"text-align: center;\">[latex]\\begin{align}f(0.5)&amp;=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &amp;=0.5\\tan(\\frac{\\pi}{4})\\\\ &amp;=0.5 \\end{align}[\/latex]<\/p>\nThis means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Figure shows the graph of one period of the function.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163810\/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\"> <b>Figure 3<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nSketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].\n\n[reveal-answer q=\"547078\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"547078\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/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.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174880[\/ohm_question]\n\n<\/div>\n<h2>Graphing One Period of a Shifted Tangent Function<\/h2>\nNow that we can graph a <strong>tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add <em>C<\/em> and <em>D<\/em> to the general form of the tangent function.\n<div>\n<div style=\"text-align: center;\">[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\nThe graph of a transformed tangent function is different from the basic tangent function tan x in several ways:\n<div class=\"textbox\"><header>\n<h3>A General Note: Features of the Graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>The range is (\u2212\u221e,\u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>There is no amplitude.<\/li>\n \t<li>[latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/h3>\n<ol>\n \t<li>Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\n \t<li>Identify the <strong>stretching\/compressing<\/strong> factor, |A|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Graphing One Period of a Shifted Tangent Function<\/h3>\nGraph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].\n\n[reveal-answer q=\"385350\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"385350\"]\n\n<strong>Step 1.<\/strong> The function is already written in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].\n\n<strong>Step 2.<\/strong>&nbsp;[latex]A=\u22122[\/latex], so the stretching factor is [latex]|A|=2[\/latex].\n\n<strong>Step 3.<\/strong>&nbsp;[latex]B=\\pi[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\pi}=1[\/latex].\n\n<strong>Step 4.<\/strong>&nbsp;[latex]C=\u2212\\pi[\/latex], so the phase shift is [latex]\\dfrac{C}{B}=\\dfrac{\u2212\\pi}{\\pi}=\u22121[\/latex].\n\n<strong>Step 5\u20137.<\/strong> The asymptotes are at [latex]x=\u2212\\frac{3}{2}[\/latex] and [latex]x=\u2212\\frac{1}{2}[\/latex] and the three recommended reference points are (\u22121.25, 1), (\u22121,\u22121), and (\u22120.75, \u22123). The graph is shown in Figure 4.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163812\/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\"> <b>Figure 4<\/b>[\/caption]\n<h4>Analysis of the Solution<\/h4>\nNote that this is a decreasing function because <em>A<\/em> &lt; 0.\n\n[\/hidden-answer]<b><\/b>\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nHow would the graph in Example 2&nbsp;look different if we made <em>A<\/em> = 2 instead of \u22122?\n\n[reveal-answer q=\"560477\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"560477\"]\n\nIt would be reflected across the line [latex]y=\u22121[\/latex], becoming an increasing function.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the graph of a tangent function, identify horizontal and vertical stretches.<\/h3>\n<ol>\n \t<li>Find the period <em>P<\/em> from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\n \t<li>Write [latex]f(x)=A\\tan\\left(\\frac{\\pi}{P}x\\right)[\/latex].<\/li>\n \t<li>Determine a convenient point (<em>x<\/em>, <em>f<\/em>(<em>x<\/em>)) on the given graph and use it to determine <em>A<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Identifying the Graph of a Stretched Tangent<\/h3>\nFind a formula for the function graphed in Figure 5.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163814\/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\"> <b>Figure 5<\/b>[\/caption]\n\n[reveal-answer q=\"606896\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"606896\"]\n\nThe graph has the shape of a tangent function.\n\n<strong>Step 1.<\/strong> 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].\n\n<strong>Step 2.<\/strong> The equation must have the [latex]\\text{form}f(x)=A\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\n\n<strong>Step 3.<\/strong> To find the vertical stretch <em>A<\/em>, we can use the point (2,2).\n<p style=\"text-align: center;\">[latex]2=A\\tan\\left(\\frac{\\pi}{8}\\times2\\right)=A\\tan\\left(\\frac{\\pi}{4}\\right)[\/latex]<\/p>\nBecause [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], <em>A<\/em> = 2.\n\nThis function would have a formula [latex]f(x)=2\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nFind a formula for the function in Figure 6.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163816\/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\"> <b>Figure 6<\/b>[\/caption]\n\n[reveal-answer q=\"359527\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"359527\"]\n\n[latex]g(x)=4\\tan(2x)[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]129731[\/ohm_question]\n\n<\/div>\n<div>\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\nMany 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 tangent function .\n<div class=\"textbox shaded\">\n<h3>Example 4: Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\nSuppose the function [latex]y=5\\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 <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.\n<ol>\n \t<li>Find and interpret the stretching factor and period.<\/li>\n \t<li>Graph on the interval [0, 5].<\/li>\n \t<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\n<\/ol>\n[reveal-answer q=\"351813\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"351813\"]\n<ol>\n \t<li>We know from the general form of &nbsp;[latex]y=A\\tan(Bt)\\\\[\/latex] &nbsp;that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.\n<figure id=\"Image_06_02_022\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163819\/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\"> <b>Figure 7<\/b>[\/caption]<\/figure>\nWe see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.\n\nThe period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\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.<\/li>\n \t<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period. See Figure 8.\n<figure id=\"Image_06_02_021\" class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163821\/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\"> <b>Figure 8<\/b>[\/caption]<\/figure>\n<\/li>\n \t<li>period: [latex]f(1)=5\\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.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\nhttps:\/\/youtu.be\/ssjG9kE25OY\n<h2>Analyzing the Graphs of y = sec x and y = cscx and Their Variations<\/h2>\nThe <strong>secant<\/strong> was defined by the <strong>reciprocal identity<\/strong>&nbsp;[latex]\\sec x=\\frac{1}{\\cos x}[\/latex]. Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}\\text{, etc}[\/latex].&nbsp;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.\n\nWe can graph [latex]y=\\sec x[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 9. 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 <strong>secant function<\/strong> 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.\n\nThe secant graph has vertical asymptotes at each value of <em>x<\/em> 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.\n\nNote that, because cosine is an even function, secant is also an even function. That is, [latex]\\sec(\u2212x)=\\sec x[\/latex].\n<figure id=\"Figure_06_02_008\" class=\"small ui-has-child-figcaption\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163823\/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.\">\n<div style=\"text-align: center;\"><strong>Figure 9.&nbsp;<\/strong>Graph of the secant function, [latex]f(x)=\\sec x=\\frac{1}{\\cos x}[\/latex]<\/div><\/figure>\nAs we did for the tangent function, we will again refer to the constant |<em>A<\/em>| as the stretching factor, not the amplitude.\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne \\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{\\pi}{2|B|}k [\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>There is no amplitude.<\/li>\n \t<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\nSimilar to the secant, the <strong>cosecant<\/strong> is defined by the reciprocal identity [latex]\\csc x=1\\sin x[\/latex]. Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, \u03c0, 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.\n\nWe can graph [latex]y=\\csc x[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 10. 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 <strong>cosecant function<\/strong> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.\n\nThe cosecant graph has vertical asymptotes at each value of <em>x<\/em> where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.\n\nNote that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\csc(\u2212x)=\u2212\\csc x[\/latex].\n\nThe graph of cosecant, which is shown in Figure 10, is similar to the graph of secant.\n<figure id=\"Figure_06_02_009\" class=\"small ui-has-child-figcaption\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163825\/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.\">\n<div style=\"text-align: center;\"><strong>Figure 10.&nbsp;<\/strong>The graph of the cosecant function, [latex]f(x)=\\csc x=\\frac{1}{\\sin x}\/latex]<\/div><\/figure>\n<div class=\"textbox\"><header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx)<\/h3>\n<\/header>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>The range is ( \u2212\u221e, \u2212|A|] \u222a [|A|, \u221e).<\/li>\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y&nbsp;<\/em>= csc <em>x<\/em><\/h2>\nFor 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 <strong>cosecant function<\/strong> in the same way as for the secant and other functions. The equations become the following.\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div style=\"text-align: center;\">[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/div>\n<div class=\"textbox\"><header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne \\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n \t<li>There is no amplitude.<\/li>\n \t<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\"><header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>There is no amplitude.<\/li>\n \t<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\sec(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function given in the form [latex]y=A\\sec(Bx)[\/latex].<\/li>\n \t<li>Identify the stretching\/compressing factor, |A|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>Sketch the graph of [latex]y=A\\cos(Bx)[\/latex].<\/li>\n \t<li>Use the reciprocal relationship between [latex]y=\\cos x[\/latex] and [latex]y=\\sec x[\/latex] to draw the graph of [latex]y=A\\sec(Bx)[\/latex].<\/li>\n \t<li>Sketch the asymptotes.<\/li>\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Graphing a Variation of the Secant Function<\/h3>\nGraph one period of [latex]f(x)=2.5\\sec(0.4x)[\/latex].\n\n[reveal-answer q=\"926159\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"926159\"]\n\n<strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\sec(Bx)[\/latex].\n<strong>Step 2.<\/strong>&nbsp;[latex]A=2.5[\/latex] so the stretching factor is 2.5.\n<strong>Step 3.<\/strong>&nbsp;[latex]B=0.4[\/latex], so [latex]P=\\frac{2\\pi}{0.4}=5\\pi[\/latex]. The period is 5\u03c0 units.\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=2.5\\cos(0.4x)[\/latex].\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.\n<strong>Steps 6\u20137.<\/strong> Sketch two asymptotes at [latex]x=1.25\\pi[\/latex]&nbsp;and [latex]x=3.75\\pi[\/latex]. We can use two reference points, the local minimum at (0, 2.5) and the local maximum at (2.5\u03c0, \u22122.5). Figure 11 shows the graph.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163829\/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\"> <b>Figure 11<\/b>[\/caption]\n\n[\/hidden-answer]<b><\/b>\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nGraph one period of [latex]f(x)=\u22122.5\\sec(0.4x)[\/latex].\n\n[reveal-answer q=\"945046\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"945046\"]\n\nThis is a vertical reflection of the preceding graph because <em>A<\/em> is negative.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163919\/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.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/h3>\n<em>Yes. The range of<\/em>&nbsp;[latex]f(x) = A\\sec(Bx \u2212 C) + D[\/latex] is ( \u2212\u221e, \u2212|<em>A<\/em>| + <em>D<\/em>] \u222a [|<em>A<\/em>| + <em>D<\/em>, \u221e).\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\sec (Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function given in the form [latex]y=A\\sec(Bx\u2212C)+D[\/latex].<\/li>\n \t<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\sec(Bx)[\/latex]. but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Graphing a Variation of the Secant Function<\/h3>\nGraph one period of [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].\n\n[reveal-answer q=\"429424\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"429424\"]\n\n<strong>Step 1.<\/strong> Express the function given in the form [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].\n\n<strong>Step 2.<\/strong> The stretching\/compressing factor is |<em>A<\/em>| = 4.\n\n<strong>Step 3.<\/strong> The period is\n<p style=\"text-align: center;\">[latex]\\begin{align} \\frac{2\\pi}{|B|}&amp;=\\frac{2\\pi}{\\frac{\\pi}{3}}\\\\ &amp;=\\frac{2\\pi}{1}\\times\\frac{3}{\\pi}\\\\ &amp;=6 \\end{align}[\/latex]<\/p>\n<strong>Step 4.<\/strong> The phase shift is\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{C}{B}&amp;=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{3}} \\\\ &amp;=\\frac{\\pi}{2} \\times \\frac{3}{\\pi} \\\\ &amp;=1.5 \\end{align}[\/latex]<\/p>\n<strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\sec(Bx)[\/latex],but shift it to the right by [latex]\\frac{C}{B}=1.5[\/latex] and up by <em>D&nbsp;<\/em>= 6.\n\n<strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x&nbsp;<\/em>= 0, <em>x<\/em> = 3, and <em>x<\/em> = 6. There is a local minimum at (1.5, 5) and a local maximum at (4.5, \u22123). Figure 12 shows the graph.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163831\/CNX_Precalc_Figure_06_02_012-1.jpg\" alt=\"\" width=\"487\" height=\"318\"> <b>Figure 12<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nGraph one period of [latex]f(x)=\u22126\\sec(4x+2)\u22128[\/latex].\n\n[reveal-answer q=\"142167\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"142167\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163922\/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.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174885[\/ohm_question]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>The domain of [latex]\\csc x[\/latex] was given to be all <em>x<\/em> such that [latex]x\\ne k\\pi[\/latex] for any integer <em>k<\/em>. Would the domain of&nbsp;[latex]y=A\\csc(Bx\u2212C)+D[\/latex] be [latex]x\\ne\\frac{C+k\\pi}{B}[\/latex]?<\/h4>\n<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>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\csc(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function given in the form [latex]y=A\\csc(Bx)[\/latex].<\/li>\n \t<li>|<em>A<\/em>|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\sin(Bx)[\/latex].<\/li>\n \t<li>Use the reciprocal relationship between [latex]y=\\sin x[\/latex] and [latex]y=\\csc x[\/latex] to draw the graph of [latex]y=A\\csc(Bx) [\/latex].<\/li>\n \t<li>Sketch the asymptotes.<\/li>\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Graphing a Variation of the Cosecant Function<\/h3>\nGraph one period of [latex]f(x)=\u22123\\csc(4x)[\/latex].\n\n[reveal-answer q=\"194858\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"194858\"]\n\n<strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\csc(Bx)[\/latex].\n\n<strong>Step 2. <\/strong>[latex]|A|=|\u22123|=3[\/latex], so the stretching factor is 3.\n\n<strong>Step 3.<\/strong> [latex]B=4\\text{, so}P=\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex].The period is [latex]\\frac{\\pi}{2}[\/latex] units.\n\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=\u22123\\sin(4x)[\/latex].\n\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function.\n\n<strong>Steps 6\u20137.<\/strong> Sketch three asymptotes at [latex]x=0\\text{, }x=\\frac{\\pi}{4}\\text{, and }x=\\frac{\\pi}{2}[\/latex].We can use two reference points, the local maximum at [latex]\\left(\\frac{\\pi}{8}\\text{, }\u22123\\right)[\/latex] and the local minimum at [latex]\\left(\\frac{3\\pi}{8}\\text{, }3\\right)[\/latex]. Figure 13 shows the graph.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163833\/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\"> <b>Figure 13<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nGraph one period of [latex]f(x)=0.5\\csc(2x)[\/latex].\n\n[reveal-answer q=\"267711\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"267711\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163924\/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.\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function given in the form [latex]y=A\\csc(Bx\u2212C)+D[\/latex].<\/li>\n \t<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n \t<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it to the right by and up by <em>D<\/em>.<\/li>\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\nSketch a graph of [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex]. What are the domain and range of this function?\n\n[reveal-answer q=\"993272\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"993272\"]\n\n<strong>Step 1.<\/strong> Express the function given in the form [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex].\n\n<strong>Step 2.<\/strong> Identify the stretching\/compressing factor, [latex]|A|=2[\/latex].\n\n<strong>Step 3.<\/strong> The period is [latex]\\frac{2\\pi}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}=\\frac{2\\pi}{1}\\times \\frac{2}{\\pi}=4[\/latex].\n\n<strong>Step 4.<\/strong> The phase shift is [latex]\\frac{0}{\\frac{\\pi}{2}}=0[\/latex].\n\n<strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it up [latex]D=1[\/latex].\n\n<strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x<\/em> = 0, <em>x<\/em> = 2, <em>x<\/em> = 4.\n\nThe graph for this function is shown in Figure 14.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163835\/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\"> <b>Figure 14<\/b>[\/caption]\n<h4>Analysis of the Solution<\/h4>\nThe 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(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)+1[\/latex], shown as the orange dashed wave.\n\n[\/hidden-answer]<b><\/b>\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nGiven the graph of [latex]f(x)=2\\cos\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] shown in Figure 15, sketch the graph of [latex]g(x)=2\\sec\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] on the same axes.\n\n&nbsp;\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163838\/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\"> <b>Figure 15<\/b>[\/caption]\n[reveal-answer q=\"560894\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"560894\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163927\/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.\">[\/hidden-answer]<\/div>\n<h2>Analyzing the Graph of y = cot x and Its Variations<\/h2>\nThe last trigonometric function we need to explore is <strong>cotangent<\/strong>. The cotangent is defined by the <strong>reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the <strong>cotangent function<\/strong> is also all real numbers.\n\nWe can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 16. 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.\n\nThe cotangent graph has vertical asymptotes at each value of <em>x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of <em>x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.\n<figure id=\"Figure_06_02_017\" class=\"small ui-has-child-figcaption\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163840\/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\"> <b>Figure 16.<\/b> The cotangent function[\/caption]\n\n<figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>The range is (\u2212\u221e, \u221e).<\/li>\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h2>\nWe can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\n<ul>\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\n \t<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>There is no amplitude.<\/li>\n \t<li>[latex]y=A\\cot(Bx)[\/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 class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\n \t<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\n \t<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex].<\/li>\n \t<li>Plot any two reference points.<\/li>\n \t<li>Use the reciprocal relationship between tangent and cotangent to draw the graph of [latex]y=A\\cot(Bx)[\/latex].<\/li>\n \t<li>Sketch the asymptotes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Graphing Variations of the Cotangent Function<\/h3>\nDetermine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.\n\n[reveal-answer q=\"32362\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"32362\"]\n\n<strong>Step 1.<\/strong> Expressing the function in the form [latex]f(x)=A\\cot(Bx)[\/latex] gives [latex]f(x)=3\\cot(4x)[\/latex].\n\n<strong>Step 2.<\/strong> The stretching factor is [latex]|A|=3[\/latex].\n\n<strong>Step 3.<\/strong> The period is [latex]P=\\frac{\\pi}{4}[\/latex].\n\n<strong>Step 4.<\/strong> Sketch the graph of [latex]y=3\\tan(4x)[\/latex].\n\n<strong>Step 5.<\/strong> Plot two reference points. Two such points are [latex]\\left(\\frac{\\pi}{16}\\text{, }3\\right)[\/latex] and [latex]\\left(\\frac{3\\pi}{16}\\text{, }\u22123\\right)[\/latex].\n\n<strong>Step 6.<\/strong> Use the reciprocal relationship to draw [latex]y=3\\cot(4x)[\/latex].\n\n<strong>Step 7.<\/strong> Sketch the asymptotes, [latex]x=0[\/latex], [latex]x=\\frac{\\pi}{4}[\/latex].\n\nThe orange graph in Figure 17 shows [latex]y=3\\tan(4x)[\/latex] and the blue graph shows [latex]y=3\\cot(4x)[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163843\/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\"> <b>Figure 17<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n \t<li>Express the function in the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/li>\n \t<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\n \t<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n \t<li>Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n \t<li>Sketch the asymptotes [latex]x =\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 11: Graphing a Modified Cotangent<\/h3>\nSketch a graph of one period of the function [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\n\n[reveal-answer q=\"706245\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"706245\"]\n\n<strong>Step 1.<\/strong> The function is already written in the general form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].\n\n<strong>Step 2.<\/strong>&nbsp;[latex]A=4[\/latex], so the stretching factor is 4.\n\n<strong>Step 3.<\/strong>&nbsp;[latex]B=\\frac{\\pi}{8}[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\frac{\\pi}{8}}=8[\/latex].\n\n<strong>Step 4.<\/strong>&nbsp;[latex]C=\\frac{\\pi}{2}[\/latex], so the phase shift is [latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{8}}=4[\/latex].\n\n<strong>Step 5.<\/strong> We draw [latex]f(x)=4\\tan\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\n\n<strong>Step 6-7.<\/strong> Three points we can use to guide the graph are (6,2), (8,\u22122), and (10,\u22126). We use the reciprocal relationship of tangent and cotangent to draw [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\n\n<strong>Step 8.<\/strong> The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=12[\/latex].\n\nThe graph is shown in Figure 18.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163846\/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\"> <b>Figure 18.<\/b> One period of a modified cotangent function.[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\n<td>[latex]y=A\\tan(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\n<td>[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cosecant<\/td>\n<td>[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\n<td>[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The tangent function has period \u03c0.<\/li>\n \t<li>[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex] is a tangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n \t<li>The secant and cosecant are both periodic functions with a period of2\u03c0. [latex]f(x)=A\\sec(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched secant function graph.<\/li>\n \t<li>[latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched cosecant function graph.<\/li>\n \t<li>The cotangent function has period \u03c0 and vertical asymptotes at 0, \u00b1\u03c0,\u00b12\u03c0,....<\/li>\n \t<li>The range of cotangent is (\u2212\u221e,\u221e),and the function is decreasing at each point in its range.<\/li>\n \t<li>The cotangent is zero at [latex]\\pm\\frac{\\pi}{2}\\text{, }\\pm\\frac{3\\pi}{2}[\/latex],....<\/li>\n \t<li>[latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex] is a cotangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n \t<li>Real-world scenarios can be solved using graphs of trigonometric functions.<\/li>\n<\/ul>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\">Analyze the graph of \u2009y=tan\u2009x and y=cot x.<\/li>\n<li style=\"font-weight: 400;\">Graph variations of \u2009y=tan\u2009x and y=cot x.<\/li>\n<li>Determine a function formula from a tangent or cotangent graph.<\/li>\n<li style=\"font-weight: 400;\">Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li style=\"font-weight: 400;\">Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li>Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\n<p>We will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\n<\/div>\n<p>The <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\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>We can determine whether tangent is an odd or even function by using the definition of tangent.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan(\u2212x)&=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} && \\text{Definition of tangent.} \\\\ &=\\frac{\u2212\\sin x}{\\cos x} && \\text{Sine is an odd function, cosine is even.} \\\\ &=\u2212\\frac{\\sin x}{\\cos x} && \\text{The quotient of an odd and an even function is odd.} \\\\ &=\u2212\\tan x && \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\n<p>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 the table below.<\/p>\n<table id=\"Table_06_02_00\" style=\"width: 1035px;\" 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 style=\"width: 30px;\"><em><strong> x <\/strong><\/em><\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30px;\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">\u20131<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">1<\/td>\n<td style=\"width: 80px;\">[latex]\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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 x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.<\/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><em><strong> x <\/strong><\/em><\/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> tan <em>x <\/em><\/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>As <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.<\/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><em><strong> x <\/strong><\/em><\/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> tan <em>x <\/em><\/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>We can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. Figure 1&nbsp;represents the graph of [latex]y=\\tan x[\/latex]. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.<\/p>\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163804\/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\"><b>Figure 1.<\/b> Graph of the tangent function<\/p>\n<\/div>\n<\/figure>\n<h2>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h2>\n<p>As with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\n<\/div>\n<p>We can identify horizontal and vertical stretches and compressions using values of A and B. 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>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 A.<\/p>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>| .<\/li>\n<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> 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}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\n<p>We can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> 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](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\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>\n<div style=\"text-align: center;\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\n<\/div>\n<p>because &nbsp;[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Identify the stretching factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Draw vertical asymptotes at &nbsp;[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\n<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\n<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]&nbsp;(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Sketching a Compressed Tangent<\/h3>\n<p>Sketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q302986\">Show Solution<\/span><\/p>\n<div id=\"q302986\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we identify <em>A<\/em> and B.<\/p>\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\/3675\/2018\/09\/27163807\/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.\" width=\"487\" height=\"113\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>Because [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(0.5)&=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &=0.5\\tan(\\frac{\\pi}{4})\\\\ &=0.5 \\end{align}[\/latex]<\/p>\n<p>This means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Figure shows the graph of one period of the function.<\/p>\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\/3675\/2018\/09\/27163810\/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\"><b>Figure 3<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q547078\">Show Solution<\/span><\/p>\n<div id=\"q547078\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/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.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174880\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174880&theme=oea&iframe_resize_id=ohm174880\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing One Period of a Shifted Tangent Function<\/h2>\n<p>Now that we can graph a <strong>tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add <em>C<\/em> and <em>D<\/em> to the general form of the tangent function.<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<p>The graph of a transformed tangent function is different from the basic tangent function tan x in several ways:<\/p>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e,\u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the <strong>stretching\/compressing<\/strong> factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> 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 class=\"textbox shaded\">\n<h3>Example 2: Graphing One Period of a Shifted Tangent Function<\/h3>\n<p>Graph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385350\">Show Solution<\/span><\/p>\n<div id=\"q385350\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The function is already written in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/p>\n<p><strong>Step 2.<\/strong>&nbsp;[latex]A=\u22122[\/latex], so the stretching factor is [latex]|A|=2[\/latex].<\/p>\n<p><strong>Step 3.<\/strong>&nbsp;[latex]B=\\pi[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\pi}=1[\/latex].<\/p>\n<p><strong>Step 4.<\/strong>&nbsp;[latex]C=\u2212\\pi[\/latex], so the phase shift is [latex]\\dfrac{C}{B}=\\dfrac{\u2212\\pi}{\\pi}=\u22121[\/latex].<\/p>\n<p><strong>Step 5\u20137.<\/strong> The asymptotes are at [latex]x=\u2212\\frac{3}{2}[\/latex] and [latex]x=\u2212\\frac{1}{2}[\/latex] and the three recommended reference points are (\u22121.25, 1), (\u22121,\u22121), and (\u22120.75, \u22123). The graph is shown in Figure 4.<\/p>\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\/3675\/2018\/09\/27163812\/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\"><b>Figure 4<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that this is a decreasing function because <em>A<\/em> &lt; 0.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>How would the graph in Example 2&nbsp;look different if we made <em>A<\/em> = 2 instead of \u22122?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q560477\">Show Solution<\/span><\/p>\n<div id=\"q560477\" class=\"hidden-answer\" style=\"display: none\">\n<p>It would be reflected across the line [latex]y=\u22121[\/latex], becoming an increasing function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the graph of a tangent function, identify horizontal and vertical stretches.<\/h3>\n<ol>\n<li>Find the period <em>P<\/em> from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\n<li>Write [latex]f(x)=A\\tan\\left(\\frac{\\pi}{P}x\\right)[\/latex].<\/li>\n<li>Determine a convenient point (<em>x<\/em>, <em>f<\/em>(<em>x<\/em>)) on the given graph and use it to determine <em>A<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Identifying the Graph of a Stretched Tangent<\/h3>\n<p>Find a formula for the function graphed in Figure 5.<\/p>\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\/3675\/2018\/09\/27163814\/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\"><b>Figure 5<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q606896\">Show Solution<\/span><\/p>\n<div id=\"q606896\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has the shape of a tangent function.<\/p>\n<p><strong>Step 1.<\/strong> 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].<\/p>\n<p><strong>Step 2.<\/strong> The equation must have the [latex]\\text{form}f(x)=A\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> To find the vertical stretch <em>A<\/em>, we can use the point (2,2).<\/p>\n<p style=\"text-align: center;\">[latex]2=A\\tan\\left(\\frac{\\pi}{8}\\times2\\right)=A\\tan\\left(\\frac{\\pi}{4}\\right)[\/latex]<\/p>\n<p>Because [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], <em>A<\/em> = 2.<\/p>\n<p>This function would have a formula [latex]f(x)=2\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find a formula for the function in Figure 6.<\/p>\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\/3675\/2018\/09\/27163816\/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\"><b>Figure 6<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q359527\">Show Solution<\/span><\/p>\n<div id=\"q359527\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=4\\tan(2x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm129731\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129731&theme=oea&iframe_resize_id=ohm129731\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div>\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\n<p>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 tangent function .<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 4: Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\n<p>Suppose the function [latex]y=5\\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 <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.<\/p>\n<ol>\n<li>Find and interpret the stretching factor and period.<\/li>\n<li>Graph on the interval [0, 5].<\/li>\n<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q351813\">Show Solution<\/span><\/p>\n<div id=\"q351813\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We know from the general form of &nbsp;[latex]y=A\\tan(Bt)\\\\[\/latex] &nbsp;that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.<br \/>\n<figure 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\/3675\/2018\/09\/27163819\/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\"><b>Figure 7<\/b><\/p>\n<\/div>\n<\/figure>\n<p>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>The period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\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.<\/li>\n<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period. See Figure 8.<br \/>\n<figure 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\/3675\/2018\/09\/27163821\/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\"><b>Figure 8<\/b><\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>period: [latex]f(1)=5\\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.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Animation:  Graphing the Tangent Function Using the Unit Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ssjG9kE25OY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Analyzing the Graphs of y = sec x and y = cscx and Their Variations<\/h2>\n<p>The <strong>secant<\/strong> was defined by the <strong>reciprocal identity<\/strong>&nbsp;[latex]\\sec x=\\frac{1}{\\cos x}[\/latex]. Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}\\text{, etc}[\/latex].&nbsp;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>We can graph [latex]y=\\sec x[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 9. 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 <strong>secant function<\/strong> 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>The secant graph has vertical asymptotes at each value of <em>x<\/em> 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>Note that, because cosine is an even function, secant is also an even function. That is, [latex]\\sec(\u2212x)=\\sec x[\/latex].<\/p>\n<figure id=\"Figure_06_02_008\" class=\"small ui-has-child-figcaption\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163823\/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.\" \/><\/p>\n<div style=\"text-align: center;\"><strong>Figure 9.&nbsp;<\/strong>Graph of the secant function, [latex]f(x)=\\sec x=\\frac{1}{\\cos x}[\/latex]<\/div>\n<\/figure>\n<p>As we did for the tangent function, we will again refer to the constant |<em>A<\/em>| as the stretching factor, not the amplitude.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an odd integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<p>Similar to the secant, the <strong>cosecant<\/strong> is defined by the reciprocal identity [latex]\\csc x=1\\sin x[\/latex]. Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, \u03c0, 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>We can graph [latex]y=\\csc x[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 10. 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 <strong>cosecant function<\/strong> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\n<p>The cosecant graph has vertical asymptotes at each value of <em>x<\/em> where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\n<p>Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\csc(\u2212x)=\u2212\\csc x[\/latex].<\/p>\n<p>The graph of cosecant, which is shown in Figure 10, is similar to the graph of secant.<\/p>\n<figure id=\"Figure_06_02_009\" class=\"small ui-has-child-figcaption\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163825\/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.\" \/><\/p>\n<div style=\"text-align: center;\"><strong>Figure 10.&nbsp;<\/strong>The graph of the cosecant function, [latex]f(x)=\\csc x=\\frac{1}{\\sin x}\/latex]<\/div>\n<\/figure>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx)<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an integer.<\/li>\n<li>The range is ( \u2212\u221e, \u2212|A|] \u222a [|A|, \u221e).<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y&nbsp;<\/em>= csc <em>x<\/em><\/h2>\n<p>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 <strong>cosecant function<\/strong> in the same way as for the secant and other functions. The equations become the following.<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div style=\"text-align: center;\">[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/div>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an odd integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\sec(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\sec(Bx)[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Sketch the graph of [latex]y=A\\cos(Bx)[\/latex].<\/li>\n<li>Use the reciprocal relationship between [latex]y=\\cos x[\/latex] and [latex]y=\\sec x[\/latex] to draw the graph of [latex]y=A\\sec(Bx)[\/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 class=\"textbox shaded\">\n<h3>Example 6: Graphing a Variation of the Secant Function<\/h3>\n<p>Graph one period of [latex]f(x)=2.5\\sec(0.4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q926159\">Show Solution<\/span><\/p>\n<div id=\"q926159\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\sec(Bx)[\/latex].<br \/>\n<strong>Step 2.<\/strong>&nbsp;[latex]A=2.5[\/latex] so the stretching factor is 2.5.<br \/>\n<strong>Step 3.<\/strong>&nbsp;[latex]B=0.4[\/latex], so [latex]P=\\frac{2\\pi}{0.4}=5\\pi[\/latex]. The period is 5\u03c0 units.<br \/>\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=2.5\\cos(0.4x)[\/latex].<br \/>\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.<br \/>\n<strong>Steps 6\u20137.<\/strong> Sketch two asymptotes at [latex]x=1.25\\pi[\/latex]&nbsp;and [latex]x=3.75\\pi[\/latex]. We can use two reference points, the local minimum at (0, 2.5) and the local maximum at (2.5\u03c0, \u22122.5). Figure 11 shows the graph.<\/p>\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\/3675\/2018\/09\/27163829\/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\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=\u22122.5\\sec(0.4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945046\">Show Solution<\/span><\/p>\n<div id=\"q945046\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a vertical reflection of the preceding graph because <em>A<\/em> is negative.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163919\/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.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/h3>\n<p><em>Yes. The range of<\/em>&nbsp;[latex]f(x) = A\\sec(Bx \u2212 C) + D[\/latex] is ( \u2212\u221e, \u2212|<em>A<\/em>| + <em>D<\/em>] \u222a [|<em>A<\/em>| + <em>D<\/em>, \u221e).<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\sec (Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\sec(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\sec(Bx)[\/latex]. but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Graphing a Variation of the Secant Function<\/h3>\n<p>Graph one period of [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q429424\">Show Solution<\/span><\/p>\n<div id=\"q429424\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Express the function given in the form [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> The stretching\/compressing factor is |<em>A<\/em>| = 4.<\/p>\n<p><strong>Step 3.<\/strong> The period is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} \\frac{2\\pi}{|B|}&=\\frac{2\\pi}{\\frac{\\pi}{3}}\\\\ &=\\frac{2\\pi}{1}\\times\\frac{3}{\\pi}\\\\ &=6 \\end{align}[\/latex]<\/p>\n<p><strong>Step 4.<\/strong> The phase shift is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{C}{B}&=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{3}} \\\\ &=\\frac{\\pi}{2} \\times \\frac{3}{\\pi} \\\\ &=1.5 \\end{align}[\/latex]<\/p>\n<p><strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\sec(Bx)[\/latex],but shift it to the right by [latex]\\frac{C}{B}=1.5[\/latex] and up by <em>D&nbsp;<\/em>= 6.<\/p>\n<p><strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x&nbsp;<\/em>= 0, <em>x<\/em> = 3, and <em>x<\/em> = 6. There is a local minimum at (1.5, 5) and a local maximum at (4.5, \u22123). Figure 12 shows the graph.<\/p>\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\/3675\/2018\/09\/27163831\/CNX_Precalc_Figure_06_02_012-1.jpg\" alt=\"\" width=\"487\" height=\"318\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=\u22126\\sec(4x+2)\u22128[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q142167\">Show Solution<\/span><\/p>\n<div id=\"q142167\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163922\/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.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174885\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174885&theme=oea&iframe_resize_id=ohm174885\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>The domain of [latex]\\csc x[\/latex] was given to be all <em>x<\/em> such that [latex]x\\ne k\\pi[\/latex] for any integer <em>k<\/em>. Would the domain of&nbsp;[latex]y=A\\csc(Bx\u2212C)+D[\/latex] be [latex]x\\ne\\frac{C+k\\pi}{B}[\/latex]?<\/h4>\n<p><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 class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\csc(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\csc(Bx)[\/latex].<\/li>\n<li>|<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\sin(Bx)[\/latex].<\/li>\n<li>Use the reciprocal relationship between [latex]y=\\sin x[\/latex] and [latex]y=\\csc x[\/latex] to draw the graph of [latex]y=A\\csc(Bx)[\/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 class=\"textbox shaded\">\n<h3>Example 8: Graphing a Variation of the Cosecant Function<\/h3>\n<p>Graph one period of [latex]f(x)=\u22123\\csc(4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q194858\">Show Solution<\/span><\/p>\n<div id=\"q194858\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\csc(Bx)[\/latex].<\/p>\n<p><strong>Step 2. <\/strong>[latex]|A|=|\u22123|=3[\/latex], so the stretching factor is 3.<\/p>\n<p><strong>Step 3.<\/strong> [latex]B=4\\text{, so}P=\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex].The period is [latex]\\frac{\\pi}{2}[\/latex] units.<\/p>\n<p><strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=\u22123\\sin(4x)[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function.<\/p>\n<p><strong>Steps 6\u20137.<\/strong> Sketch three asymptotes at [latex]x=0\\text{, }x=\\frac{\\pi}{4}\\text{, and }x=\\frac{\\pi}{2}[\/latex].We can use two reference points, the local maximum at [latex]\\left(\\frac{\\pi}{8}\\text{, }\u22123\\right)[\/latex] and the local minimum at [latex]\\left(\\frac{3\\pi}{8}\\text{, }3\\right)[\/latex]. Figure 13 shows the graph.<\/p>\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\/3675\/2018\/09\/27163833\/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\"><b>Figure 13<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=0.5\\csc(2x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q267711\">Show Solution<\/span><\/p>\n<div id=\"q267711\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163924\/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.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\csc(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it to the right by and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\n<p>Sketch a graph of [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex]. What are the domain and range of this function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993272\">Show Solution<\/span><\/p>\n<div id=\"q993272\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Express the function given in the form [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> Identify the stretching\/compressing factor, [latex]|A|=2[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> The period is [latex]\\frac{2\\pi}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}=\\frac{2\\pi}{1}\\times \\frac{2}{\\pi}=4[\/latex].<\/p>\n<p><strong>Step 4.<\/strong> The phase shift is [latex]\\frac{0}{\\frac{\\pi}{2}}=0[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it up [latex]D=1[\/latex].<\/p>\n<p><strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x<\/em> = 0, <em>x<\/em> = 2, <em>x<\/em> = 4.<\/p>\n<p>The graph for this function is shown in Figure 14.<\/p>\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\/3675\/2018\/09\/27163835\/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\"><b>Figure 14<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>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(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)+1[\/latex], shown as the orange dashed wave.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given the graph of [latex]f(x)=2\\cos\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] shown in Figure 15, sketch the graph of [latex]g(x)=2\\sec\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] on the same axes.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163838\/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\"><b>Figure 15<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q560894\">Show Solution<\/span><\/p>\n<div id=\"q560894\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163927\/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.\" \/><\/div>\n<\/div>\n<\/div>\n<h2>Analyzing the Graph of y = cot x and Its Variations<\/h2>\n<p>The last trigonometric function we need to explore is <strong>cotangent<\/strong>. The cotangent is defined by the <strong>reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the <strong>cotangent function<\/strong> is also all real numbers.<\/p>\n<p>We can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 16. 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>The cotangent graph has vertical asymptotes at each value of <em>x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of <em>x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.<\/p>\n<figure id=\"Figure_06_02_017\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163840\/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\"><b>Figure 16.<\/b> The cotangent function<\/p>\n<\/div><figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u221e).<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h2>\n<p>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>\n<div style=\"text-align: center;\">[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/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 <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\cot(Bx)[\/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 class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\n<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\n<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\tan(Bx)[\/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\\cot(Bx)[\/latex].<\/li>\n<li>Sketch the asymptotes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Graphing Variations of the Cotangent Function<\/h3>\n<p>Determine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32362\">Show Solution<\/span><\/p>\n<div id=\"q32362\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Expressing the function in the form [latex]f(x)=A\\cot(Bx)[\/latex] gives [latex]f(x)=3\\cot(4x)[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> The stretching factor is [latex]|A|=3[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> The period is [latex]P=\\frac{\\pi}{4}[\/latex].<\/p>\n<p><strong>Step 4.<\/strong> Sketch the graph of [latex]y=3\\tan(4x)[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Plot two reference points. Two such points are [latex]\\left(\\frac{\\pi}{16}\\text{, }3\\right)[\/latex] and [latex]\\left(\\frac{3\\pi}{16}\\text{, }\u22123\\right)[\/latex].<\/p>\n<p><strong>Step 6.<\/strong> Use the reciprocal relationship to draw [latex]y=3\\cot(4x)[\/latex].<\/p>\n<p><strong>Step 7.<\/strong> Sketch the asymptotes, [latex]x=0[\/latex], [latex]x=\\frac{\\pi}{4}[\/latex].<\/p>\n<p>The orange graph in Figure 17 shows [latex]y=3\\tan(4x)[\/latex] and the blue graph shows [latex]y=3\\cot(4x)[\/latex].<\/p>\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\/3675\/2018\/09\/27163843\/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\"><b>Figure 17<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function in the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching factor, |<em>A<\/em>|.<\/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\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the asymptotes [latex]x =\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> 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 class=\"textbox shaded\">\n<h3>Example 11: Graphing a Modified Cotangent<\/h3>\n<p>Sketch a graph of one period of the function [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706245\">Show Solution<\/span><\/p>\n<div id=\"q706245\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The function is already written in the general form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/p>\n<p><strong>Step 2.<\/strong>&nbsp;[latex]A=4[\/latex], so the stretching factor is 4.<\/p>\n<p><strong>Step 3.<\/strong>&nbsp;[latex]B=\\frac{\\pi}{8}[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\frac{\\pi}{8}}=8[\/latex].<\/p>\n<p><strong>Step 4.<\/strong>&nbsp;[latex]C=\\frac{\\pi}{2}[\/latex], so the phase shift is [latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{8}}=4[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> We draw [latex]f(x)=4\\tan\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<p><strong>Step 6-7.<\/strong> Three points we can use to guide the graph are (6,2), (8,\u22122), and (10,\u22126). We use the reciprocal relationship of tangent and cotangent to draw [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<p><strong>Step 8.<\/strong> The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=12[\/latex].<\/p>\n<p>The graph is shown in Figure 18.<\/p>\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\/3675\/2018\/09\/27163846\/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\"><b>Figure 18.<\/b> One period of a modified cotangent function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\n<td>[latex]y=A\\tan(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\n<td>[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cosecant<\/td>\n<td>[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\n<td>[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The tangent function has period \u03c0.<\/li>\n<li>[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex] is a tangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n<li>The secant and cosecant are both periodic functions with a period of2\u03c0. [latex]f(x)=A\\sec(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched secant function graph.<\/li>\n<li>[latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched cosecant function graph.<\/li>\n<li>The cotangent function has period \u03c0 and vertical asymptotes at 0, \u00b1\u03c0,\u00b12\u03c0,....<\/li>\n<li>The range of cotangent is (\u2212\u221e,\u221e),and the function is decreasing at each point in its range.<\/li>\n<li>The cotangent is zero at [latex]\\pm\\frac{\\pi}{2}\\text{, }\\pm\\frac{3\\pi}{2}[\/latex],....<\/li>\n<li>[latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex] is a cotangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n<li>Real-world scenarios can be solved using graphs of trigonometric functions.<\/li>\n<\/ul>\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-1377\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Project<\/strong>: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Animation: Graphing the Tangent Function Using the Unit Circle. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ssjG9kE25OY\">https:\/\/youtu.be\/ssjG9kE25OY<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/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":503070,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Animation: Graphing the Tangent Function Using the Unit Circle\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ssjG9kE25OY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1377","chapter","type-chapter","status-publish","hentry"],"part":1375,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1377","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1377\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1375"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1377\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1377"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1377"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1377"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}