{"id":2990,"date":"2019-10-30T17:18:53","date_gmt":"2019-10-30T17:18:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=2990"},"modified":"2025-03-31T19:03:58","modified_gmt":"2025-03-31T19:03:58","slug":"3-5-the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/3-5-the-other-trigonometric-functions\/","title":{"raw":"3.5 The Other Trigonometric Functions","rendered":"3.5 The Other Trigonometric Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of [latex]\\frac{\\pi }{3},\\text{ }\\frac{\\pi }{4},\\text{ and }\\frac{\\pi }{6}.[\/latex]<\/li>\r\n \t<li>Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent.<\/li>\r\n \t<li>Use properties of even and odd trigonometric functions.<\/li>\r\n \t<li>Recognize and use fundamental identities.<\/li>\r\n \t<li>Evaluate trigonometric functions with a calculator.<\/li>\r\n \t<li>Describe the graphical properties of the other trigonometric functions.<\/li>\r\n \t<li>Sketch the tangent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137640206\">A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is [latex]\\frac{1}{12}[\/latex] or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions.\u00a0 Though sine and cosine are the trigonometric functions most often used, we know from our work with right triangles that there are six trigonometric functions altogether. In this section, we will investigate the remaining functions in terms of using ideas from the unit circle.<\/p>\r\n\r\n<div id=\"fs-id1165135437156\" class=\"bc-section section\">\r\n<h3>Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent<\/h3>\r\nRecall the following information that was covered in Section 3.1.\r\n\r\nConsider a right triangle \u25b3 ABC, with the right angle at C and with lengths a, b, and c, as in the Figure 1 below. For the acute angle A, call the leg BC its <strong>opposite side<\/strong>, and call the leg AC its <strong>adjacent side<\/strong>. Recall that the <strong>hypotenuse<\/strong> of the triangle is always opposite the right angle.\u00a0 In the triangle below, this is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we define the <strong>six trigonometric functions<\/strong> of A as follows:\r\n\r\n[caption id=\"attachment_1704\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1704 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/25220212\/Screen-Shot-2019-03-25-at-6.01.54-PM-300x269.png\" alt=\"Sides of a right triangle with respect to angle A. Right triangle with hypotenuse, opposite and adjacent sides labeled.\" width=\"300\" height=\"269\" \/> <strong>Figure 1:\u00a0\u00a0<\/strong>Sides of a right triangle with respect to angle A.[\/caption]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\"><caption>Table 1 The six trigonometric functions of A<\/caption>\r\n<thead>\r\n<tr>\r\n<td>Name of function<\/td>\r\n<td>Abbreviation<\/td>\r\n<td>Definition<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>sine(<em>A)<\/em><\/td>\r\n<td>sin(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{opposite side}}{\\text{hypotenuse}}$<\/td>\r\n<td>$latex =\\frac{a}{c}$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>cosine(<em>A)<\/em><\/td>\r\n<td>cos(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{adjacent side}}{\\text{hypotenuse}}$<\/td>\r\n<td>$latex =\\frac{b}{c}$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>tangent(<em>A)<\/em><\/td>\r\n<td>tan(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{opposite side}}{\\text{adjacent side}}$<\/td>\r\n<td>$latex =\\frac{a}{b}$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>cosecant(<em>A)<\/em><\/td>\r\n<td>csc(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{hypotenuse}}{\\text{opposite side}}$<\/td>\r\n<td>$latex =\\frac{c}{a}$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>secant(<em>A)<\/em><\/td>\r\n<td>sec(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{hypotenuse}}{\\text{adjacent side}}$<\/td>\r\n<td>$latex =\\frac{c}{b}$<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>cotangent(<em>A)<\/em><\/td>\r\n<td>cot(<em>A)<\/em><\/td>\r\n<td>$latex =\\frac{\\text{adjacent side}}{\\text{opposite side}}$<\/td>\r\n<td>$latex =\\frac{b}{a}$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe will usually use the abbreviated names of the functions. Notice from Table 1 that the pairs sin(A) and csc(A), cos(A) and sec(A), and tan(A) and cot(A) are reciprocals:\r\n<table style=\"border-collapse: collapse; width: 100%; height: 24px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\csc \\left(A\\right) =\\frac{1}{\\sin\\left(A\\right)}$<\/td>\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\sec\\left(A\\right) =\\frac{1}{\\cos\\left(A\\right)}$<\/td>\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\cot\\left(A\\right) =\\frac{1}{\\tan\\left(A\\right)}$<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\sin\\left(A\\right) =\\frac{1}{\\csc\\left(A\\right)}$<\/td>\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\cos\\left(A\\right) =\\frac{1}{\\sec\\left(A\\right)}$<\/td>\r\n<td style=\"width: 33.3333%; height: 12px;\">$latex \\tan\\left(A\\right) =\\frac{1}{\\cot\\left(A\\right)}$<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"font-size: 1rem; text-align: initial;\">Also recall the work we did in section 3.3 when we defined the sine and cosine functions in terms of the unit circle.\u00a0 For any angle [latex]t,[\/latex] we labeled the intersection of the terminal side and the unit circle as by its coordinates, [latex]\\left(x,y\\right).[\/latex] \u00a0 We considered an acute angle in the first quadrant and dropped a perpendicular line to the x- axis to create a right triangle. The sides of the right triangle were then [latex]x[\/latex] and [latex]y.[\/latex]\u00a0 \u00a0When we used our right trigonometric definitions above, we saw that [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{1}[\/latex] and [latex]\\mathrm{sin}\\left(t\\right)=\\frac{y}{1}.[\/latex]\u00a0 This means the ordered pair [latex]\\left(x,y\\right)=\\left(\\mathrm{cos}\\left(t\\right),\\mathrm{sin}\\left(t\\right)\\right).[\/latex]\u00a0 See Figure 2 below.<\/span>\r\n\r\nAs with the sine and cosine, we can use the [latex]\\left(x,y\\right)[\/latex] coordinates to find the other functions.\r\n<div id=\"Figure_05_02_002\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"403\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132221\/CNX_Precalc_Figure_05_02_003.jpg\" alt=\"Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t. The terminal side of the angle intersects the circle at the point (x,y), which is equivalent to (cos t, sin t).\" width=\"403\" height=\"187\" \/> <strong>Figure 2:<\/strong> Unit circle where the central angle is in radians[\/caption]\r\n\r\n<\/div>\r\n<div id=\"Figure_05_03_001\" class=\"small\"><\/div>\r\n<ul>\r\n \t<li id=\"fs-id1165137426281\">\u00a0In right triangle trigonometry, the tangent of an angle is the ratio of the opposite side over the adjacent side with respect to the angle.\u00a0 In <a class=\"autogenerated-content\" href=\"#Figure_05_03_001\">Figure 2<\/a>, the tangent of angle [latex]t[\/latex] is equal to [latex]\\frac{y}{x},\\text{ where }x\\ne0.[\/latex]<\/li>\r\n \t<li>Because the y-value is equal to the sine of [latex]t,[\/latex] and the <em>x<\/em>-value is equal to the cosine of [latex]t,[\/latex] the tangent of angle [latex]t[\/latex] can also be defined as [latex]\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)},\\text{ where }\\mathrm{cos}\\left(t\\right)\\ne0.[\/latex]<\/li>\r\n \t<li>The remaining three functions can also all be expressed as functions of a point on the unit circle.<\/li>\r\n \t<li>When we change\u00a0the y-value to the sine of [latex]t,[\/latex] and the <em>x<\/em>-value to the cosine of [latex]t,[\/latex] we can\u00a0 express the functions in terms of the sine and cosine functions.\u00a0 When we do this, we typically refer to these statements as <strong>basic trignometric identities<\/strong>.\u00a0 \u00a0 See the definition box below for details.<\/li>\r\n<\/ul>\r\n<div id=\"fs-id1165137580858\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<h3>Tangent, Secant, Cosecant, and Cotangent Functions and Basic Identities<\/h3>\r\n<p id=\"eip-id1165137834409\">If [latex]t[\/latex] is a real number and [latex]\\left(x,y\\right)[\/latex] is a point where the terminal side of an angle of [latex]t[\/latex] radians intercepts the unit circle, then we can create the equations below and their corresponding identities since\u00a0we know that [latex]x=\\mathrm{cos}\\left(t\\right)\\text{ and }y=\\mathrm{sin}\\left(t\\right)[\/latex].<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 47px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 9px;\">\r\n<td style=\"width: 30%; text-align: left; height: 9px;\">Definition<\/td>\r\n<td style=\"width: 70%; text-align: left; height: 9px;\">Trigonometric Identity<\/td>\r\n<\/tr>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"width: 30%; height: 10px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x},\\text{ }x\\ne 0[\/latex]<\/td>\r\n<td style=\"width: 70%; height: 10px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)},\\text{ }\\mathrm{cos}\\left(t\\right)\\ne 0 [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 9px;\">\r\n<td style=\"width: 30%; height: 9px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x},\\text{ }x\\ne0 [\/latex]<\/td>\r\n<td style=\"width: 70%; height: 9px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)},\\text{ }\\mathrm{cos}\\left(t\\right)\\ne 0 [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 9px;\">\r\n<td style=\"width: 30%; height: 10px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y},\\text{ }y\\ne0 [\/latex]<\/td>\r\n<td style=\"width: 70%; height: 10px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)},\\text{ }\\mathrm{sin}\\left(t\\right)\\ne 0 [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 9px;\">\r\n<td style=\"width: 30%; height: 9px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y},\\text{ }y\\ne0 [\/latex]<\/td>\r\n<td style=\"width: 70%; height: 9px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)},\\text{ }\\mathrm{sin}\\left(t\\right)\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div id=\"Example_05_03_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137572568\">\r\n<div id=\"fs-id1165137723147\">\r\n<h3>Example 1:\u00a0 Finding Trigonometric Functions from a Point on the Unit Circle<\/h3>\r\n<p id=\"fs-id1165134122782\" style=\"text-align: left;\">The point [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\frac{1}{2}\\right)[\/latex] is on the unit circle, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_03_002\">\u00a0Figure 3<\/a>.\u00a0 \u00a0Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_03_002\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"401\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132415\/CNX_Precalc_Figure_05_03_002.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1\/2) is at intersection of terminal side of angle and edge of circle.\" width=\"401\" height=\"178\" \/> <strong>Figure 3:<\/strong>\u00a0 Graph of circle with angle of t inscribed.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135241178\">[reveal-answer q=\"fs-id1165135241178\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135241178\"]\r\n<p id=\"fs-id1165135571770\">Because we know the [latex]\\left(x,y\\right)[\/latex] coordinates of the point on the unit circle indicated by angle [latex]t,[\/latex] we can use those coordinates to find the six functions.\u00a0 First we know<\/p>\r\n\r\n<div id=\"fs-id1165137482825\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{sin}\\left(t\\right)&amp;=y=\\frac{1}{2}\\\\ \\mathrm{cos}\\left(t\\right)&amp;=x=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>Since [latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x}[\/latex] or [latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}[\/latex] using the values for sine and cosine, we have<\/div>\r\n<div style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x}=\\frac{\\frac{1}{2}}{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}=\\frac{1}{2}\\left(-\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\right)=-\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div>Since [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x}[\/latex] or [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)}[\/latex] and we know the value for cosine, we have<\/div>\r\n<div style=\"text-align: center;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}=-\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}=-\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div>Since\u00a0 [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y}[\/latex] or [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)}[\/latex] and we know the value for sine, we have<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\frac{1}{2}}=2.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">\r\n<div><\/div>\r\n<div style=\"text-align: left;\">Finally, since [latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y}[\/latex] or [latex]\\mathrm{cot}\\left(t\\right)=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}[\/latex] using the values for sine and cosine, we have<\/div>\r\n<div style=\"text-align: center;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}{\\frac{1}{2}}=\\frac{-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\left(\\frac{2}{1}\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137722103\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_05_03_01\">\r\n<div id=\"fs-id1165132947401\">\r\n\r\nThe point [latex]\\left(\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}\\right)[\/latex] is on the unit circle, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_003\">\u00a0Figure 4<\/a>.\r\n<p id=\"fs-id1165134122782\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_05_03_002\" class=\"small\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<div class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"401\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132418\/CNX_Precalc_Figure_05_03_003.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.\" width=\"401\" height=\"286\" \/> <strong>Figure 4:<\/strong>\u00a0 Graph of circle with angle of t inscribed.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137471776\">[reveal-answer q=\"fs-id1165137471776\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137471776\"]\r\n<p id=\"fs-id1165137803942\">[latex]\\mathrm{sin}\\left(t\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{cos}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{tan}\\left(t\\right)=-1,\\text{ }\\mathrm{sec}\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{csc}\\left(t\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{cot}\\left(t\\right)=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137766765\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_05_03_08\">\r\n<div id=\"fs-id1165137793763\">\r\n<p id=\"fs-id1165137793764\">Find the values of the six trigonometric functions of angle [latex]t[\/latex] based on <a class=\"autogenerated-content\" href=\"#Figure_05_03_010\">Figure 5<\/a><strong>. <\/strong><\/p>\r\n\r\n<div id=\"Figure_05_03_010\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132437\/CNX_Precalc_Figure_05_03_010.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"406\" \/> <strong>Figure 5:<\/strong>\u00a0 Graph of circle with angle of t inscribed.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135640611\">[reveal-answer q=\"fs-id1165135640611\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135640611\"]\r\n<p id=\"fs-id1165135640612\">[latex]\\begin{align*}\\mathrm{sin}\\left(t\\right)&amp;=-1,\\\\\\mathrm{cos}\\left(t\\right)&amp;=0,\\\\\\mathrm{tan}\\left(t\\right)&amp;=\\text{Undefined}\\\\ \\mathrm{sec}\\left(t\\right)&amp;=\\text{ Undefined,}\\\\\\mathrm{csc}\\left(t\\right)&amp;=-1,\\\\\\mathrm{cot}\\left(t\\right)&amp;=0\\end{align*}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_05_03_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134094606\">\r\n<div id=\"fs-id1165137629063\">\r\n<h3>Example 2: Finding the Trigonometric Functions of an Angle<\/h3>\r\n<p id=\"fs-id1165137612022\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right)[\/latex] when [latex]t=\\frac{\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135508641\">[reveal-answer q=\"fs-id1165135508641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135508641\"]\r\n<p id=\"fs-id1165137641007\">We have previously used the relationship of the sides of our special right triangles to demonstrate that [latex]\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex] and [latex]\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}.[\/latex] We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)}{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)}=\\frac{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=1[\/latex][latex]\\\\[\/latex]<\/p>\r\n\r\n<div><\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{sec}\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)} =\\frac{1}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}=\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{csc}\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)}=\\frac{1}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}=\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{cot}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)}{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)} =\\frac{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=1[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135496597\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_05_03_02\">\r\n<div id=\"fs-id1165135203430\">\r\n<p id=\"fs-id1165135203431\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right)[\/latex] when [latex]t=\\frac{\\pi }{3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132988446\">[reveal-answer q=\"fs-id1165132988446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132988446\"]\r\n<p id=\"fs-id1165132988447\">[latex]\\begin{align*}\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)&amp;=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\\\ \\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)&amp;=\\frac{1}{2}\\\\ \\mathrm{tan}\\left(\\frac{\\pi }{3}\\right)&amp;=\\sqrt[\\leftroot{1}\\uproot{2} ]{3}\\\\ \\mathrm{sec}\\left(\\frac{\\pi }{3}\\right)&amp;=2\\\\ \\mathrm{csc}\\left(\\frac{\\pi }{3}\\right)&amp;=\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}\\\\ \\mathrm{cot}\\left(\\frac{\\pi }{3}\\right)&amp;=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}\\end{align*}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137598924\">Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting [latex]x[\/latex] equal to the cosine and [latex]y[\/latex] equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in <a class=\"autogenerated-content\" href=\"#Table_05_03_01\">Table 2<\/a>.<\/p>\r\n\r\n<table id=\"Table_05_03_01\" style=\"width: 1030px;\" summary=\"..\"><caption><strong>Table 2<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Angle<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\"><strong> [latex]0[\/latex] <\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\"><strong>[latex]\\frac{\\pi }{6},\\text{ or 30\u00b0}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\"><strong>[latex]\\frac{\\pi }{4},\\text{ or 45\u00b0}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\"><strong>[latex]\\frac{\\pi }{3},\\text{ or 60\u00b0}[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\"><strong>[latex]\\frac{\\pi }{2},\\text{ or 90\u00b0}[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cosine<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 86.0781px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Sine<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Tangent<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Secant<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cosecant<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">Undefined<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cotangent<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">Undefined<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134284490\" class=\"bc-section section\">\r\n<div id=\"fs-id1321556\" class=\"bc-section section\">\r\n<h3>Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent<\/h3>\r\n<p id=\"fs-id1165137664668\">We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the <span class=\"no-emphasis\">reference angle<\/span> formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by <em>x\u00a0<\/em>and <em>y<\/em>-values in the original quadrant. <a class=\"autogenerated-content\" href=\"#Figure_05_03_004\">\u00a0Figure 6<\/a> shows which functions are positive in which quadrant.<\/p>\r\n<p id=\"fs-id1165137663666\">To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase \u201cA Smart Trig Class.\u201d Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is \u201c<strong>A<\/strong>,\u201d <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, \u201c<strong>S<\/strong>mart,\u201d only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, \u201c<strong>T<\/strong>rig,\u201d only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, \u201c<strong>C<\/strong>lass,\u201d only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.<\/p>\r\n\r\n<div id=\"Figure_05_03_004\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"398\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132421\/CNX_Precalc_Figure_05_03_004.jpg\" alt=\"Graph of circle with each quadrant labeled. Under quadrant 1, labels fro sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant 2, labels for sin t and csc t. Under quadrant 3, labels for tan t and cot t. Under quadrant 4, labels for cos t, sec t.\" width=\"398\" height=\"297\" \/> <strong>Figure 6:<\/strong> Quadrants with trig functions that are positive.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134468910\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137863545\"><strong>Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135485968\" type=\"1\">\r\n \t<li>If the angle is not between [latex]0[\/latex] and [latex]2\\pi[\/latex] or [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex], first add or subtract as many full revolutions as necessary in order to find a coterminal angle that is within these boundaries.<\/li>\r\n \t<li>Measure the angle formed by the terminal side of your angle and the horizontal axis. This is the reference angle.<\/li>\r\n \t<li>Evaluate the function at the reference angle.<\/li>\r\n \t<li>Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_03_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137735785\">\r\n<div id=\"fs-id1165135168424\">\r\n<h3>Example 3: Using Reference Angles to Find Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1165133004515\">Use reference angles to find all six trigonometric functions of [latex]-\\frac{5\\pi }{6}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135704070\">[reveal-answer q=\"fs-id1165135704070\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135704070\"]\r\n<p id=\"fs-id1165135704072\">A coterminal angle within 0 and [latex]2\\pi[\/latex] will be [latex]\\frac{7\\pi}{6}.[\/latex] The angle between this angle\u2019s terminal side and the <em>x<\/em>-axis is [latex]\\frac{\\pi }{6},[\/latex] so that is the reference angle. Since [latex]-\\frac{5\\pi }{6}[\/latex] is in the third quadrant, where both [latex]x[\/latex] and [latex]y[\/latex] are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.<\/p>\r\n\r\n<div id=\"eip-id1165135700065\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{cos}\\left(-\\frac{5\\pi }{6}\\right)&amp;=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\\\\\mathrm{sin}\\left(-\\frac{5\\pi }{6}\\right)&amp;=-\\frac{1}{2},\\\\\\mathrm{tan}\\left(-\\frac{5\\pi }{6}\\right)&amp;=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3} \\\\ \\mathrm{sec}\\left(-\\frac{5\\pi }{6}\\right)&amp;=-\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3},\\\\\\mathrm{csc}\\left(-\\frac{5\\pi }{6}\\right)&amp;=-2,\\\\\\mathrm{cot}\\left(-\\frac{5\\pi }{6}\\right)&amp;=\\sqrt[\\leftroot{1}\\uproot{2} ]{3} \\end{align*}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137627037\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_05_03_03\">\r\n<div id=\"fs-id1165137414778\">\r\n<p id=\"fs-id1165137476436\">Use reference angles to find all six trigonometric functions of [latex]-\\frac{7\\pi }{4}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135196912\">[reveal-answer q=\"fs-id1165135196912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135196912\"] A co-terminal angle is [latex]\\frac{\\pi}{4}[\/latex] and since this angle is in quadrant 1, it is also the reference angle.\r\n<p id=\"fs-id1165135196914\">[latex]\\mathrm{sin}\\left(\\frac{-7\\pi }{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{cos}\\left(\\frac{-7\\pi }{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{tan}\\left(\\frac{-7\\pi }{4}\\right)=1,[\/latex]<\/p>\r\n\r\n<div><\/div>\r\n[latex]\\mathrm{sec}\\left(\\frac{-7\\pi }{4}\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{csc}\\left(\\frac{-7\\pi }{4}\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{cot}\\left(\\frac{-7\\pi }{4}\\right)=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Using Even and Odd Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1165137571788\">To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.<\/p>\r\nRecall that:\r\n<ul>\r\n \t<li id=\"fs-id1165134042164\">An <span class=\"no-emphasis\">even function<\/span> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right).[\/latex]<\/li>\r\n \t<li id=\"fs-id1165137443964\">An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\r\n<\/ul>\r\n<span style=\"font-size: 1rem; text-align: initial;\">We can test whether a trigonometric function is even or odd by drawing a <\/span><span class=\"no-emphasis\" style=\"font-size: 1rem; text-align: initial;\">unit circle<\/span><span style=\"font-size: 1rem; text-align: initial;\"> with a positive and a negative angle, as in <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Figure_05_03_007\">\u00a0Figure 7<\/a><span style=\"font-size: 1rem; text-align: initial;\">. The sine of the positive angle is [latex]y.[\/latex] The sine of the negative angle is \u2212<\/span><em style=\"font-size: 1rem; text-align: initial;\">y<\/em><span style=\"font-size: 1rem; text-align: initial;\">. The <\/span><span class=\"no-emphasis\" style=\"font-size: 1rem; text-align: initial;\">sine function<\/span><span style=\"font-size: 1rem; text-align: initial;\">, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Table_05_03_02\">Table 3<\/a><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<div id=\"Figure_05_03_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"395\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132429\/CNX_Precalc_Figure_05_03_007.jpg\" alt=\"Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.\" width=\"395\" height=\"299\" \/> <strong>Figure 7:<\/strong>\u00a0 Graph of circle with angle of t and -t inscribed.[\/caption]\r\n\r\n<\/div>\r\n<table id=\"Table_05_03_02\" class=\"aligncenter\" style=\"width: 769px;\" summary=\"..\"><caption><strong>Table 3<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{sin}\\text{ }\\left(t\\right)&amp;=y\\\\ \\mathrm{sin}\\left(-t\\right)&amp;=-y\\\\ \\mathrm{sin}\\left(-t\\right)&amp;=\\mathrm{-sin}\\text{} \\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{cos}\\text{ }\\left(t\\right)&amp;=x\\\\\\mathrm{cos}\\left(-t\\right)&amp;=x\\\\ \\mathrm{cos}\\left(-t\\right)&amp;=\\mathrm{cos}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{tan}\\left(t\\right)&amp;=\\frac{y}{x}\\\\\\mathrm{tan}\\left(-t\\right)&amp;=-\\frac{y}{x} \\\\ \\mathrm{tan}\\left(-t\\right)&amp;=\\mathrm{-tan}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{csc}\\text{ }\\left(t\\right)&amp;=\\frac{1}{y}\\\\\\mathrm{csc}\\left(-t\\right)&amp;=\\frac{1}{-y}\\\\ \\mathrm{csc}\\left(-t\\right)&amp;=\\mathrm{-csc}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{sec}\\text{ }\\left(t\\right)&amp;=\\frac{1}{x}\\\\\\mathrm{sec}\\left(-t\\right)&amp;=\\frac{1}{x}\\\\ \\mathrm{sec}\\left(-t\\right)&amp;=\\mathrm{sec}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{cot}\\text{ }\\left(t\\right)&amp;=\\frac{x}{y}\\\\\\mathrm{cot}\\left(-t\\right)&amp;=\\frac{x}{-y}\\\\ \\mathrm{cot}\\left(-t\\right)&amp;=\\mathrm{-cot}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137824115\">\r\n<p id=\"fs-id1165137456021\">Therefore, we can see that cosine and secant are even and sine, tangent, cosecant, and cotangent are odd.<\/p>\r\n\r\n<div id=\"Example_05_03_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137559306\">\r\n<div id=\"fs-id1165137559308\">\r\n<h3>Example 4: Using Even and Odd Properties of Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1165137736069\">If the secant of angle [latex]t[\/latex] is 2, what is the secant of [latex]-t?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134043531\">[reveal-answer q=\"fs-id1165134043531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134043531\"]\r\n<p id=\"fs-id1165134043534\">Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle <em>t<\/em> is 2, the secant of [latex]-t[\/latex] is also 2.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137697003\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"ti_05_03_04\">\r\n<div id=\"fs-id1165135251458\">\r\n<p id=\"fs-id1165135251459\">If the cotangent of angle [latex]t[\/latex] is [latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3},[\/latex] what is the cotangent of [latex]-t?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137807232\">[reveal-answer q=\"fs-id1165137807232\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137807232\"]\r\n<p id=\"fs-id1165137807233\">[latex]-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137404616\" class=\"bc-section section\">\r\n<div id=\"fs-id1165134267806\" class=\"bc-section section\">\r\n<h3>Recognizing and Using Fundamental Identities<\/h3>\r\n<p id=\"fs-id1165137655585\">We have now explored a number of\u00a0 definitions and properties of trigonometric functions and can use them to help us find values for other trigonometric function values for a specific angle.\u00a0 We can also use the definitions to help simplify trigonometric expressions.\u00a0 As you continue on to Calculus, you will see that it is oftentimes advantageous to work with the simplest expression possible.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137404616\" class=\"bc-section section\">\r\n<div id=\"fs-id1165134267806\" class=\"bc-section section\">\r\n<div id=\"Example_05_03_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135414324\">\r\n<div id=\"fs-id1165135414326\">\r\n<h3>Example 5: Using Identities to Simplify Trigonometric Expressions<\/h3>\r\n<p id=\"fs-id1165137433430\">Simplify [latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137456723\">[reveal-answer q=\"fs-id1165137456723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137456723\"]\r\n<p id=\"fs-id1165137471602\">We can simplify this by rewriting both functions in terms of sine and cosine.<\/p>\r\n\r\n<div id=\"eip-id1165133223704\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}&amp;=\\frac{\\frac{1}{\\mathrm{cos}\\left(t\\right)}}{\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}}&amp;&amp;\\text{ }\\\\\\text{ }&amp;=\\frac{1}{\\mathrm{cos}\\left(t\\right)}\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}&amp;&amp;\\text{To divide the functions, multiply by the reciprocal}.\\\\\\text{ }&amp;=\\frac{1}{\\mathrm{sin}\\left(t\\right)}&amp;&amp;\\text{Divide out the cosines}.\\\\\\text{ }&amp;=\\mathrm{csc}\\left(t\\right)&amp;&amp;\\text{Simplify and use the identity. } \\end{align*}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165135496299\">By showing that [latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}[\/latex] can be simplified to [latex]\\mathrm{csc}\\left(t\\right),[\/latex] we have, in fact, established a new identity.<\/p>\r\n\r\n<div id=\"eip-id1165137404775\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}=\\mathrm{csc}\\left(t\\right).[\/latex]<\/div>\r\n<h3>Analysis<\/h3>\r\n<div id=\"fs-id1165135414324\">\r\n<div id=\"fs-id1165137456723\">\r\n<div>When simplifying trigonometric expressions, we need to consider the domain restrictions of the original expression before any algebra is done.\u00a0 The equivalent expression will only be valid on the original expression's domain.\u00a0 The expression\u00a0[latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}[\/latex] has restrictions which include those of the domain of the secant function, the domain of the tangent function and finally those where the tangent function is zero.<\/div>\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1165135414324\">\r\n<div id=\"fs-id1165137456723\">\r\n<div>\r\n\r\nThe domain of the tangent and secant function both exclude values where [latex]cos\\left(t\\right) = 0[\/latex] or where [latex]t= \\frac{\\pi}{2}[\/latex] or [latex]t= \\frac{3\\pi}{2}[\/latex] and any coterminal angles.\u00a0 This list of exceptions is often written as [latex]\\frac{\\pi}{2}+ \\pi k[\/latex] where [latex]k[\/latex] is an integer.\u00a0 Further, tangent is zero where [latex]sin\\left(t\\right)=0[\/latex] or where [latex]t=0[\/latex] or [latex]\\pi[\/latex] and any coterminal angles.\u00a0 These domain exceptions can be written as [latex]t = k \\pi[\/latex] where [latex]k[\/latex] is an integer.\u00a0 Therefore, our final equality\r\n<div id=\"fs-id1165135414324\">\r\n<div id=\"fs-id1165137456723\">\r\n<div id=\"eip-id1165137404775\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}=\\mathrm{csc}\\left(t\\right)\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\nis only valid where [latex]sin\\left(t\\right)\\ne0[\/latex] and [latex]cos\\left(t\\right)\\ne0[\/latex] or where [latex]t\\ne\\frac{\\pi}{2}k[\/latex] for [latex]k[\/latex] an integer.\r\n\r\n<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137936650\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"ti_05_03_06\">\r\n<div id=\"fs-id1165137659799\">\r\n<p id=\"fs-id1165137659800\">Simplify [latex]\\left(\\mathrm{tan}\\left(t\\right)\\right)\\left(\\mathrm{cos}\\left(t\\right)\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134148506\">[reveal-answer q=\"fs-id1165134148506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134148506\"]\r\n<p id=\"fs-id1165134148508\">[latex]\\mathrm{sin}\\left(t\\right)[\/latex]<\/p>\r\nThe domain of the tangent function excludes values where [latex]cos\\left(t\\right) = 0[\/latex] or where [latex]t= \\frac{\\pi}{2}[\/latex] or [latex]t= \\frac{3\\pi}{2}[\/latex] and any coterminal angles.\u00a0 This list of exceptions is often written as [latex]\\frac{\\pi}{2}+ \\pi k[\/latex] where [latex]k[\/latex] is an integer.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137732163\" class=\"bc-section section\">\r\n<h4>The Pythagorean Identity<\/h4>\r\n<p id=\"fs-id1165135203709\"><span style=\"font-size: 1rem; text-align: initial;\">You should recall that as a direct result of our definition of the sine and cosine functions in terms of the coordinates of points on the unit circle, we were able to create the Pythagorean Identity given below:\u00a0 \u00a0<\/span><\/p>\r\n<p style=\"text-align: center;\">[latex]{\\mathrm{cos}}^{2}\\left(\\theta\\right)+{\\mathrm{sin}}^{2}\\left(\\theta\\right)=1.[\/latex]<\/p>\r\nThis identity can often be used to find the sine or cosine function value if we know one of these values for a given angle.\u00a0 By using the other basic identities, we can then find the values of all of the trigonometric functions for a given angle.\r\n<div id=\"Example_05_03_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137863436\">\r\n<div id=\"fs-id1165137863439\">\r\n<h3>Example 6: Using Identities to Relate Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1165135516891\">If [latex]\\text{cos}\\left(t\\right)=\\frac{12}{13}[\/latex] and [latex]t[\/latex] is in quadrant IV, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_03_008\">\u00a0Figure 8<\/a>, find the values of the other five trigonometric functions.<\/p>\r\n\r\n<div id=\"Figure_05_03_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"381\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132432\/CNX_Precalc_Figure_05_03_008.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (12\/13, y) is at intersection of terminal side of angle and edge of circle.\" width=\"381\" height=\"300\" \/> <strong>Figure 8:<\/strong>\u00a0 Graph of circle with angle of t inscribed.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137589991\">[reveal-answer q=\"fs-id1165137589991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137589991\"]\r\n<p id=\"fs-id1165135485151\">We can find the sine using the Pythagorean Identity, [latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1,[\/latex] and the remaining functions by relating them to sine and cosine.<\/p>\r\n\r\n<div id=\"eip-id1165135191762\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}{\\left(\\frac{12}{13}\\right)}^{2}+\\mathrm{sin}^{2}\\left(t\\right)&amp;=1 \\\\\\mathrm{sin}^{2}\\left(t\\right)&amp;=1-{\\left(\\frac{12}{13}\\right)}^{2}\\\\ \\mathrm{sin}^{2}\\left(t\\right)&amp;=1-\\frac{144}{169}\\\\ \\mathrm{sin}^{2}\\left(t\\right)&amp;=\\frac{25}{169}\\\\\\mathrm{sin}\\left(t\\right)&amp;=\\pm\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{25}{169}}\\\\\\mathrm{sin}\\left(t\\right)&amp;=\\pm\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{25}}{\\sqrt[\\leftroot{1}\\uproot{2} ]{169}}\\\\ \\mathrm{sin}\\left(t\\right)&amp;=\\pm\\frac{5}{13} \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135333148\">The sign of the sine depends on the <em>y<\/em>-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the <em>y<\/em>-values are negative, its sine is negative, [latex]-\\frac{5}{13}.[\/latex]<\/p>\r\n<p id=\"fs-id1165135403285\">The remaining functions can be calculated using identities relating them to sine and cosine.<\/p>\r\n\r\n<div id=\"fs-id1165137705474\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{tan}\\left(t\\right)&amp;=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}=\\frac{-\\frac{5}{13}}{\\frac{12}{13}}=-\\frac{5}{12} \\\\ \\mathrm{sec}\\left(t\\right)&amp;=\\frac{1}{\\mathrm{cos}\\left(t\\right)}=\\frac{1}{\\frac{12}{13}}=\\frac{13}{12}\\\\ \\mathrm{csc}\\left(t\\right)&amp;=\\frac{1}{\\mathrm{sin}\\left(t\\right)}=\\frac{1}{-\\frac{5}{13}}=-\\frac{13}{5}\\\\\\mathrm{cot}\\left(t\\right)&amp;=\\frac{1}{\\mathrm{tan}\\left(t\\right)}=\\frac{1}{-\\frac{5}{12}}=-\\frac{12}{5}\\end{align*}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137564622\" class=\"precalculus tryit\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"ti_05_03_07\">\r\n<div id=\"fs-id1165135570427\">\r\n<p id=\"fs-id1165137855086\">If [latex]\\mathrm{sec}\\left(t\\right)=-\\frac{17}{8}[\/latex] and [latex]0\\lt t&lt;\\pi ,[\/latex] \u00a0find the values of the other five functions.\u00a0 Hint:\u00a0 First find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] using a reciprocal relationship.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137426588\">[reveal-answer q=\"fs-id1165137426588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137426588\"]\r\n<p id=\"fs-id1165137426589\">[latex]\\mathrm{cos}\\left(t\\right)=-\\frac{8}{17},\\text{ }\\mathrm{sin}\\left(t\\right)=\\frac{15}{17},\\text{ }\\mathrm{tan}\\left(t\\right)=-\\frac{15}{8}[\/latex]<\/p>\r\n\r\n<div><\/div>\r\n[latex]\\mathrm{csc}\\left(t\\right)=\\frac{17}{15},\\text{ }\\mathrm{cot}\\left(t\\right)=-\\frac{8}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137934400\" class=\"bc-section section\">\r\n<h3>Evaluating Trigonometric Functions with a Calculator<\/h3>\r\n<p id=\"fs-id1165135530385\">We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.<\/p>\r\n<p id=\"fs-id1165135555490\">Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.<\/p>\r\n<p id=\"fs-id1165137704595\">If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor [latex]\\frac{\\pi }{180}[\/latex] to convert the degrees to radians. To find the secant of [latex]30\u00b0,[\/latex] we could press<\/p>\r\n\r\n<div id=\"eip-887\" class=\"unnumbered\">(for a scientific calculator): [latex]\\frac{1}{30\u00d7\\frac{\\pi }{180}}\\text{COS}[\/latex]<\/div>\r\n<p id=\"fs-id1165135342785\">or<\/p>\r\n\r\n<div id=\"eip-148\" class=\"unnumbered\">(for a graphing calculator):\u00a0 [latex]\\frac{1}{\\mathrm{cos}\\left(\\frac{30\\pi }{180}\\right)}[\/latex]<\/div>\r\n<div id=\"fs-id1165135208418\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137758027\"><strong>Given an angle measure in radians, use a <span style=\"text-decoration: underline;\">scientific<\/span> calculator to find the cosecant. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137758031\" type=\"1\">\r\n \t<li>If the calculator has degree mode and radian mode, set it to radian mode.<\/li>\r\n \t<li>Enter: [latex]1\\text{ \/}[\/latex]<\/li>\r\n \t<li>Enter the value of the angle inside parentheses.<\/li>\r\n \t<li>Press the SIN key.<\/li>\r\n \t<li>Press the = key.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137597190\"><strong>Given an angle measure in radians, use a <span style=\"text-decoration: underline;\">graphing utility<\/span>\/calculator to find the cosecant. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137470238\" type=\"1\">\r\n \t<li>If the graphing utility has degree mode and radian mode, set it to radian mode.<\/li>\r\n \t<li>Enter: [latex]1\\text{ \/}[\/latex]<\/li>\r\n \t<li>Press the SIN key.<\/li>\r\n \t<li>Enter the value of the angle inside parentheses.<\/li>\r\n \t<li>Press the ENTER key.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_05_03_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137572560\">\r\n<div id=\"fs-id1165135380123\">\r\n<h3>Example 7:\u00a0 Evaluating the Cosecant Using Technology<\/h3>\r\n<p id=\"fs-id1165135380128\">Evaluate the cosecant of [latex]\\frac{5\\pi }{7}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135698644\">[reveal-answer q=\"fs-id1165135698644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135698644\"]\r\n<p id=\"fs-id1165135698646\">For a scientific calculator, enter information as follows:<\/p>\r\n\r\n<div id=\"fs-id1165135698649\" class=\"unnumbered\">[latex]\\text{1 \/ ( 5 }\u00d7\\text{ }\\pi \\text{ \/ 7 ) SIN =}[\/latex]<\/div>\r\n<div id=\"eip-id1165133337484\" class=\"unnumbered\">[latex]\\mathrm{csc}\\left(\\frac{5\\pi }{7}\\right)\\approx 1.279[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div>For a graphing calculator, enter the information as follows:<\/div>\r\n<div>[latex]1\/[\/latex] and now press the SIN key.\u00a0 Then enter [latex]\\frac{5\\pi}{7}[\/latex] followed by a closing parenthesis. Now press ENTER.<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1165135698644\">\r\n<div><\/div>\r\n<div class=\"unnumbered\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134216886\" class=\"precalculus tryit\">\r\n<h3>Try it #8<\/h3>\r\n<div id=\"ti_05_03_10\">\r\n<div id=\"fs-id1165134237307\">\r\n<p id=\"fs-id1165134237308\">Evaluate the cotangent of [latex]-\\frac{\\pi }{8}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137435028\">[reveal-answer q=\"fs-id1165137435028\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137435028\"]\r\n<p id=\"fs-id1165137435029\">[latex]\\approx -2.414[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135593133\" class=\"precalculus media\">\r\n<div id=\"fs-id1165134284490\" class=\"bc-section section\">\r\n<div id=\"fs-id1321556\" class=\"bc-section section\">\r\n<h3>Analyzing the Graph of <em>y<\/em> = tan(<em>x)<\/em><\/h3>\r\n<p id=\"fs-id1530665\">We will begin with the graph of the <span class=\"no-emphasis\">tangent<\/span> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\r\n\r\n<div id=\"fs-id1614888\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(x\\right)=\\frac{\\mathrm{sin}\\left(x\\right)}{\\mathrm{cos}\\left(x\\right)}.[\/latex][latex]\\\\[\/latex]<\/div>\r\nRemember that there are some values of [latex]x[\/latex] for which [latex]\\mathrm{cos}\\left(x\\right)=0.[\/latex] For example, [latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0[\/latex] and [latex]\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0.[\/latex] At these values, the <span class=\"no-emphasis\">tangent function<\/span> is undefined, so the graph of [latex]y=\\mathrm{tan}\\left(x\\right)[\/latex] has discontinuities at [latex]x=\\frac{\\pi }{2}[\/latex] and [latex]\\frac{3\\pi }{2}.[\/latex] We will examine the function from a numerical point of view to see if there is evidence that there are vertical asymptotes at these points of discontinuity.\r\n<p id=\"fs-id1647294\">We have already shown, using the ideas from the unit circle, that the tangent function is odd.<\/p>\r\n<p id=\"fs-id1460636\">We can further analyze the numerical behavior of the tangent function by looking at values for some of the special angles, as listed in <a class=\"autogenerated-content\" href=\"#Table_06_02_00\">Table 4<\/a>.<\/p>\r\n\r\n<table id=\"Table_06_02_00\" style=\"height: 94px;\" 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).\"><caption><strong>Table 4<\/strong><\/caption>\r\n<tbody>\r\n<tr style=\"height: 54px;\">\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 40px;\">\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\"><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">undefined<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">\u20131<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70px; text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1279869\">These points will help us draw our graph, but we need to determine how the graph behaves where the function 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 1.5707,[\/latex] we will evaluate [latex]x[\/latex] at radian measures [latex]1.05 \\lt x \\lt 1.5707[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Table_06_02_01\">Table 5<\/a>.<\/p>\r\n\r\n<table id=\"Table_06_02_01\" style=\"width: 220px;\" 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).\"><caption><strong>Table 5<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 30px; text-align: center;\">1.3<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.5<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.55<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.56<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.57<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\"><strong>[latex]\\mathrm{tan} \\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 30px; text-align: center;\">3.6<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">14.1<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">48.1<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">92.6<\/td>\r\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1255.8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<p id=\"fs-id1658883\">As [latex]x[\/latex] approaches [latex]\\frac{\\pi }{2}[\/latex] from the left hand side, the outputs of the function get larger and larger or as [latex]x\\to{\\frac{\\pi}{2}}^{-},\\text{ }\\mathrm{tan}\\left(x\\right)\\to\\infty.[\/latex] This provides us with evidence that there is a vertical asymptote at\u00a0[latex]\\frac{\\pi }{2}.[\/latex]<\/p>\r\nBecause [latex]y=\\mathrm{tan}\\left(x\\right)[\/latex] is an odd function, we see the corresponding table of negative values in <a class=\"autogenerated-content\" href=\"#Table_06_02_02\">Table 6<\/a>.\r\n<table id=\"Table_06_02_02\" style=\"height: 73px; width: 270px;\" 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).\" width=\"457\"><caption><strong>Table 6<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 188.494px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 33.0398px; text-align: center;\">\u22121.3<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.5<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.55<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.56<\/td>\r\n<td class=\"border\" style=\"width: 18.4943px; text-align: center;\">-1.57<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 188.494px; text-align: center;\"><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 33.0398px; text-align: center;\">\u22123.6<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221214.1<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221248.1<\/td>\r\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221292.6<\/td>\r\n<td class=\"border\" style=\"width: 18.4943px; text-align: center;\">-1255.8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1318044\">We can see that, as [latex]x[\/latex] approaches [latex]-\\frac{\\pi }{2}[\/latex] from the right hand side, the outputs get more and more negative [latex]x\\to{\\frac{-\\pi}{2}}^{+},\\text{ }\\mathrm{tan}\\left(x\\right)\\to-\\infty.[\/latex]\u00a0 Again, this gives us evidence that these is a vertical asymptote at [latex]-\\frac{\\pi }{2}.[\/latex]<\/p>\r\n<a class=\"autogenerated-content\" href=\"#Figure_06_02_001\">Figure 9<\/a> represents the graph of [latex]y=\\mathrm{tan}\\left(x\\right).[\/latex] The tangent is positive from 0 to [latex]\\frac{\\pi }{2}[\/latex] and from [latex]\\pi [\/latex] to [latex]\\frac{3\\pi }{2},[\/latex] corresponding to quadrants I and III of the unit circle.\r\n\r\nWe could create more points for other intervals and we will see that the tangent function repeats its behavior every [latex]\\pi[\/latex] units.\u00a0 We therefore conclude that the tangent function has a period of\u00a0[latex]\\pi[\/latex].\r\n<div id=\"Figure_06_02_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"462\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133144\/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=\"462\" height=\"300\" \/> Figure 9: Graph of the tangent function.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<h3><\/h3>\r\n<h3>Analyzing the Graphs of <em>y<\/em> = sec(<em>x)<\/em> and <em>y<\/em> = csc(<em>x)<\/em><\/h3>\r\n<p id=\"fs-id1405395\">The <span class=\"no-emphasis\">secant<\/span> was defined by the <span class=\"no-emphasis\">reciprocal identity<\/span> [latex]\\mathrm{sec}\\left(x\\right)=\\frac{1}{\\mathrm{cos}\\left(x\\right)}.[\/latex] Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi }{2},[\/latex] [latex]\\frac{3\\pi }{2},[\/latex] etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\r\n<p id=\"fs-id1615608\">We can graph [latex]y=\\mathrm{sec}\\left(x\\right)[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_008\">Figure 10<\/a>. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <span class=\"no-emphasis\">secant function<\/span> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.<\/p>\r\n<p id=\"fs-id1657904\">The secant graph has vertical asymptotes at each value of [latex]x[\/latex] where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.<\/p>\r\n<p id=\"fs-id1457737\">Note that, because cosine is an even function, secant is also an even function. That is, [latex]\\mathrm{sec}\\left(-x\\right)=\\mathrm{sec}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_02_008\" class=\"small\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"\" align=\"aligncenter\" width=\"383\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133203\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" width=\"383\" height=\"298\" \/> Figure 10: Graph of the secant function, [latex]f\\left(x\\right)=\\mathrm{sec}\\left(x\\right)=\\frac{1}{\\mathrm{cos}\\left(x\\right).}[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1700128\">\r\n<p id=\"fs-id1380717\">Similar to the secant, the <span class=\"no-emphasis\">cosecant<\/span> is defined by the reciprocal identity [latex]\\mathrm{csc}\\left(x\\right)=\\frac{1}{\\mathrm{sin}}\\left(x\\right).[\/latex] Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at [latex]0,[\/latex] [latex]\\pi,[\/latex] etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\r\n<p id=\"fs-id1413697\">We can graph [latex]y=\\mathrm{csc}\\left(x\\right)[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">Figure 11<\/a>. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <span class=\"no-emphasis\">cosecant function<\/span> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\r\n<p id=\"fs-id1380921\">The cosecant graph has vertical asymptotes at each value of [latex]x[\/latex] where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\r\n<p id=\"fs-id1457402\">Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\mathrm{csc}\\left(-x\\right)=\\mathrm{-csc}\\left(x\\right).[\/latex]<\/p>\r\n<p id=\"eip-610\">The graph of cosecant, which is shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">Figure 9<\/a>, is similar to the graph of secant.<\/p>\r\n\r\n<div id=\"Figure_06_02_009\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"388\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133206\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" width=\"388\" height=\"300\" \/> <strong>Figure 11:<\/strong> The graph of the cosecant function, [latex]f\\left(x\\right)=\\mathrm{csc}\\left(x\\right)=\\frac{1}{\\mathrm{sin}\\left(x\\right).}[\/latex][\/caption]<\/div>\r\n<\/div>\r\n<h3>Analyzing the Graph of <em>y<\/em> = cot(<em>x)<\/em><\/h3>\r\n<p id=\"fs-id1585113\">The last trigonometric function we need to explore is <span class=\"no-emphasis\">cotangent<\/span>. The cotangent is defined by the <span class=\"no-emphasis\">reciprocal identity<\/span> [latex]\\mathrm{cot}\\left(x\\right)=\\frac{1}{\\mathrm{tan}\\left(x\\right)}.[\/latex] Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at [latex]0,\\pi ,[\/latex] etc. Since the output of the tangent function is all real numbers, the output of the <span class=\"no-emphasis\">cotangent function<\/span> is also all real numbers.<\/p>\r\n<p id=\"fs-id1333454\">We can graph [latex]y=\\mathrm{cot}\\left(x\\right)[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_017\">Figure 12<\/a>. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\r\n<p id=\"fs-id1430801\">The cotangent graph has vertical asymptotes at each value of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)=0;[\/latex] we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\mathrm{cot}\\left(x\\right)[\/latex] has vertical asymptotes at all values of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)=0,[\/latex] and [latex]\\mathrm{cot}\\left(x\\right)=0[\/latex] at all values of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)[\/latex] has its vertical asymptotes.<\/p>\r\n\r\n<div id=\"Figure_06_02_017\" class=\"small\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"332\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133237\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"332\" height=\"299\" \/> <strong>Figure 12:<\/strong> The cotangent function.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137771142\">\r\n<h3>Period of a Function<\/h3>\r\nAs we have previously discussed, a function that repeats its values in regular intervals is known as a <span class=\"no-emphasis\">periodic function<\/span>.\u00a0 For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or [latex]2\\pi ,[\/latex] will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.\r\n<p id=\"fs-id1165135593361\">Remember, the period [latex]P[\/latex] of a repeating function [latex]f[\/latex] is the number representing the interval such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex] for any value of [latex]x.[\/latex]<\/p>\r\n<p id=\"fs-id1165135333706\">The period of the cosine, sine, secant, and cosecant functions is [latex]2\\pi .[\/latex]<\/p>\r\n<p id=\"fs-id1165137893417\">The period of the tangent and cotangent functions is [latex]\\pi .[\/latex]<span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">\u00a0<\/span><\/p>\r\nOther functions can also be periodic. For example, the lengths of months repeat every four years. If [latex]x[\/latex] represents the length time, measured in years, and [latex]f\\left(x\\right)[\/latex] represents the number of days in February, then [latex]f\\left(x+4\\right)=f\\left(x\\right).[\/latex] This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A <strong>period<\/strong> is the shortest interval over which a function completes one full cycle\u2014in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135191384\">Access these online resources for additional instruction and practice with other trigonometric functions.<\/p>\r\n\r\n<ul id=\"fs-id1165137473589\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/trigfuncval\">Determining Trig Function Values<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/moretrigfun\">More Examples of Determining Trig Functions<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/pythagiden\">Pythagorean Identities<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/trigcalc\">Trig Functions on a Calculator<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137938685\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165134112952\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 116.531px;\">Tangent function<\/td>\r\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}t\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 116.531px;\">Secant function<\/td>\r\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 116.531px;\">Cosecant function<\/td>\r\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 116.531px;\">Cotangent function<\/td>\r\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{1}{\\mathrm{tan}\\left(t\\right)}=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137832791\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul>\r\n \t<li>The tangent of an angle is the ratio of the <em>y<\/em>-value to the <em>x<\/em>-value of the corresponding point on the unit circle.<\/li>\r\n \t<li>Secant, cotangent, and cosecant are all reciprocals of other functions. The secant function is the reciprocal of the cosine function, the cotangent function is the reciprocal of the tangent function, and the cosecant function is the reciprocal of the sine function.<\/li>\r\n \t<li>The six trigonometric functions can be found from a point on the unit circle.<\/li>\r\n \t<li>Trigonometric functions can also be found from an angle.<\/li>\r\n \t<li>Trigonometric functions of angles outside the first quadrant can be determined using reference angles.<\/li>\r\n \t<li>A function is said to be even if [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex] and odd if [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165134211396\">\r\n \t<li>Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.<\/li>\r\n \t<li>Even and odd properties can be used to evaluate trigonometric functions.<\/li>\r\n \t<li>The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.<\/li>\r\n \t<li>Identities can be used to evaluate trigonometric functions.<\/li>\r\n \t<li>Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities.<\/li>\r\n \t<li>The trigonometric functions repeat at regular intervals.<\/li>\r\n \t<li>The period [latex]P[\/latex] of a repeating function [latex]f[\/latex] is the smallest interval such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex] for any value of [latex]x.[\/latex]<\/li>\r\n \t<li>The values of trigonometric functions of special angles can be found by mathematical analysis.<\/li>\r\n \t<li>To evaluate trigonometric functions of other angles, we can use a calculator or computer software.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137640892\">\r\n \t<dt>cosecant<\/dt>\r\n \t<dd id=\"fs-id1165137640897\">the reciprocal of the sine function: on the unit circle, [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y},y\\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137529833\">\r\n \t<dt>cotangent<\/dt>\r\n \t<dd id=\"fs-id1165137529838\">the reciprocal of the tangent function: on the unit circle, [latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y},y\\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137410476\">\r\n \t<dt>identities<\/dt>\r\n \t<dd id=\"fs-id1165137410481\">statements that are true for all values of the input on which they are defined<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135195519\">\r\n \t<dt>period<\/dt>\r\n \t<dd id=\"fs-id1165135195524\">the smallest interval [latex]P[\/latex] of a repeating function [latex]f[\/latex] such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135591029\">\r\n \t<dt>secant<\/dt>\r\n \t<dd id=\"fs-id1165135591035\">the reciprocal of the cosine function: on the unit circle, [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x},x\\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137834349\">\r\n \t<dt>tangent<\/dt>\r\n \t<dd id=\"fs-id1165137834355\">the quotient of the sine and cosine: on the unit circle, [latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x},x\\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of [latex]\\frac{\\pi }{3},\\text{ }\\frac{\\pi }{4},\\text{ and }\\frac{\\pi }{6}.[\/latex]<\/li>\n<li>Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent.<\/li>\n<li>Use properties of even and odd trigonometric functions.<\/li>\n<li>Recognize and use fundamental identities.<\/li>\n<li>Evaluate trigonometric functions with a calculator.<\/li>\n<li>Describe the graphical properties of the other trigonometric functions.<\/li>\n<li>Sketch the tangent function.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137640206\">A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is [latex]\\frac{1}{12}[\/latex] or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions.\u00a0 Though sine and cosine are the trigonometric functions most often used, we know from our work with right triangles that there are six trigonometric functions altogether. In this section, we will investigate the remaining functions in terms of using ideas from the unit circle.<\/p>\n<div id=\"fs-id1165135437156\" class=\"bc-section section\">\n<h3>Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent<\/h3>\n<p>Recall the following information that was covered in Section 3.1.<\/p>\n<p>Consider a right triangle \u25b3 ABC, with the right angle at C and with lengths a, b, and c, as in the Figure 1 below. For the acute angle A, call the leg BC its <strong>opposite side<\/strong>, and call the leg AC its <strong>adjacent side<\/strong>. Recall that the <strong>hypotenuse<\/strong> of the triangle is always opposite the right angle.\u00a0 In the triangle below, this is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we define the <strong>six trigonometric functions<\/strong> of A as follows:<\/p>\n<div id=\"attachment_1704\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1704\" class=\"wp-image-1704 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/25220212\/Screen-Shot-2019-03-25-at-6.01.54-PM-300x269.png\" alt=\"Sides of a right triangle with respect to angle A. Right triangle with hypotenuse, opposite and adjacent sides labeled.\" width=\"300\" height=\"269\" \/><\/p>\n<p id=\"caption-attachment-1704\" class=\"wp-caption-text\"><strong>Figure 1:\u00a0\u00a0<\/strong>Sides of a right triangle with respect to angle A.<\/p>\n<\/div>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1 The six trigonometric functions of A<\/caption>\n<thead>\n<tr>\n<td>Name of function<\/td>\n<td>Abbreviation<\/td>\n<td>Definition<\/td>\n<td><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>sine(<em>A)<\/em><\/td>\n<td>sin(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{opposite side}}{\\text{hypotenuse}}[\/latex]<\/td>\n<td>[latex]=\\frac{a}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>cosine(<em>A)<\/em><\/td>\n<td>cos(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{adjacent side}}{\\text{hypotenuse}}[\/latex]<\/td>\n<td>[latex]=\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>tangent(<em>A)<\/em><\/td>\n<td>tan(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{opposite side}}{\\text{adjacent side}}[\/latex]<\/td>\n<td>[latex]=\\frac{a}{b}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>cosecant(<em>A)<\/em><\/td>\n<td>csc(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{hypotenuse}}{\\text{opposite side}}[\/latex]<\/td>\n<td>[latex]=\\frac{c}{a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>secant(<em>A)<\/em><\/td>\n<td>sec(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{hypotenuse}}{\\text{adjacent side}}[\/latex]<\/td>\n<td>[latex]=\\frac{c}{b}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>cotangent(<em>A)<\/em><\/td>\n<td>cot(<em>A)<\/em><\/td>\n<td>[latex]=\\frac{\\text{adjacent side}}{\\text{opposite side}}[\/latex]<\/td>\n<td>[latex]=\\frac{b}{a}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We will usually use the abbreviated names of the functions. Notice from Table 1 that the pairs sin(A) and csc(A), cos(A) and sec(A), and tan(A) and cot(A) are reciprocals:<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 24px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\csc \\left(A\\right) =\\frac{1}{\\sin\\left(A\\right)}[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\sec\\left(A\\right) =\\frac{1}{\\cos\\left(A\\right)}[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\cot\\left(A\\right) =\\frac{1}{\\tan\\left(A\\right)}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\sin\\left(A\\right) =\\frac{1}{\\csc\\left(A\\right)}[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\cos\\left(A\\right) =\\frac{1}{\\sec\\left(A\\right)}[\/latex]<\/td>\n<td style=\"width: 33.3333%; height: 12px;\">[latex]\\tan\\left(A\\right) =\\frac{1}{\\cot\\left(A\\right)}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Also recall the work we did in section 3.3 when we defined the sine and cosine functions in terms of the unit circle.\u00a0 For any angle [latex]t,[\/latex] we labeled the intersection of the terminal side and the unit circle as by its coordinates, [latex]\\left(x,y\\right).[\/latex] \u00a0 We considered an acute angle in the first quadrant and dropped a perpendicular line to the x- axis to create a right triangle. The sides of the right triangle were then [latex]x[\/latex] and [latex]y.[\/latex]\u00a0 \u00a0When we used our right trigonometric definitions above, we saw that [latex]\\mathrm{cos}\\left(t\\right)=\\frac{x}{1}[\/latex] and [latex]\\mathrm{sin}\\left(t\\right)=\\frac{y}{1}.[\/latex]\u00a0 This means the ordered pair [latex]\\left(x,y\\right)=\\left(\\mathrm{cos}\\left(t\\right),\\mathrm{sin}\\left(t\\right)\\right).[\/latex]\u00a0 See Figure 2 below.<\/span><\/p>\n<p>As with the sine and cosine, we can use the [latex]\\left(x,y\\right)[\/latex] coordinates to find the other functions.<\/p>\n<div id=\"Figure_05_02_002\" class=\"small\">\n<div style=\"width: 413px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132221\/CNX_Precalc_Figure_05_02_003.jpg\" alt=\"Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t. The terminal side of the angle intersects the circle at the point (x,y), which is equivalent to (cos t, sin t).\" width=\"403\" height=\"187\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2:<\/strong> Unit circle where the central angle is in radians<\/p>\n<\/div>\n<\/div>\n<div id=\"Figure_05_03_001\" class=\"small\"><\/div>\n<ul>\n<li id=\"fs-id1165137426281\">\u00a0In right triangle trigonometry, the tangent of an angle is the ratio of the opposite side over the adjacent side with respect to the angle.\u00a0 In <a class=\"autogenerated-content\" href=\"#Figure_05_03_001\">Figure 2<\/a>, the tangent of angle [latex]t[\/latex] is equal to [latex]\\frac{y}{x},\\text{ where }x\\ne0.[\/latex]<\/li>\n<li>Because the y-value is equal to the sine of [latex]t,[\/latex] and the <em>x<\/em>-value is equal to the cosine of [latex]t,[\/latex] the tangent of angle [latex]t[\/latex] can also be defined as [latex]\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)},\\text{ where }\\mathrm{cos}\\left(t\\right)\\ne0.[\/latex]<\/li>\n<li>The remaining three functions can also all be expressed as functions of a point on the unit circle.<\/li>\n<li>When we change\u00a0the y-value to the sine of [latex]t,[\/latex] and the <em>x<\/em>-value to the cosine of [latex]t,[\/latex] we can\u00a0 express the functions in terms of the sine and cosine functions.\u00a0 When we do this, we typically refer to these statements as <strong>basic trignometric identities<\/strong>.\u00a0 \u00a0 See the definition box below for details.<\/li>\n<\/ul>\n<div id=\"fs-id1165137580858\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<h3>Tangent, Secant, Cosecant, and Cotangent Functions and Basic Identities<\/h3>\n<p id=\"eip-id1165137834409\">If [latex]t[\/latex] is a real number and [latex]\\left(x,y\\right)[\/latex] is a point where the terminal side of an angle of [latex]t[\/latex] radians intercepts the unit circle, then we can create the equations below and their corresponding identities since\u00a0we know that [latex]x=\\mathrm{cos}\\left(t\\right)\\text{ and }y=\\mathrm{sin}\\left(t\\right)[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 47px;\">\n<tbody>\n<tr style=\"height: 9px;\">\n<td style=\"width: 30%; text-align: left; height: 9px;\">Definition<\/td>\n<td style=\"width: 70%; text-align: left; height: 9px;\">Trigonometric Identity<\/td>\n<\/tr>\n<tr style=\"height: 10px;\">\n<td style=\"width: 30%; height: 10px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x},\\text{ }x\\ne 0[\/latex]<\/td>\n<td style=\"width: 70%; height: 10px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)},\\text{ }\\mathrm{cos}\\left(t\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 9px;\">\n<td style=\"width: 30%; height: 9px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x},\\text{ }x\\ne0[\/latex]<\/td>\n<td style=\"width: 70%; height: 9px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)},\\text{ }\\mathrm{cos}\\left(t\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 9px;\">\n<td style=\"width: 30%; height: 10px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y},\\text{ }y\\ne0[\/latex]<\/td>\n<td style=\"width: 70%; height: 10px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)},\\text{ }\\mathrm{sin}\\left(t\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 9px;\">\n<td style=\"width: 30%; height: 9px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y},\\text{ }y\\ne0[\/latex]<\/td>\n<td style=\"width: 70%; height: 9px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)},\\text{ }\\mathrm{sin}\\left(t\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>\n<div>[latex]\\\\[\/latex]<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<div id=\"Example_05_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137572568\">\n<div id=\"fs-id1165137723147\">\n<h3>Example 1:\u00a0 Finding Trigonometric Functions from a Point on the Unit Circle<\/h3>\n<p id=\"fs-id1165134122782\" style=\"text-align: left;\">The point [latex]\\left(-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\frac{1}{2}\\right)[\/latex] is on the unit circle, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_03_002\">\u00a0Figure 3<\/a>.\u00a0 \u00a0Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_05_03_002\" class=\"small\">\n<div style=\"width: 411px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132415\/CNX_Precalc_Figure_05_03_002.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1\/2) is at intersection of terminal side of angle and edge of circle.\" width=\"401\" height=\"178\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3:<\/strong>\u00a0 Graph of circle with angle of t inscribed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241178\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135241178\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135241178\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571770\">Because we know the [latex]\\left(x,y\\right)[\/latex] coordinates of the point on the unit circle indicated by angle [latex]t,[\/latex] we can use those coordinates to find the six functions.\u00a0 First we know<\/p>\n<div id=\"fs-id1165137482825\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{sin}\\left(t\\right)&=y=\\frac{1}{2}\\\\ \\mathrm{cos}\\left(t\\right)&=x=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\end{align*}[\/latex]<\/div>\n<div><\/div>\n<div>Since [latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x}[\/latex] or [latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}[\/latex] using the values for sine and cosine, we have<\/div>\n<div style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x}=\\frac{\\frac{1}{2}}{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}=\\frac{1}{2}\\left(-\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\right)=-\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}.[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div>Since [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x}[\/latex] or [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)}[\/latex] and we know the value for cosine, we have<\/div>\n<div style=\"text-align: center;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}=-\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}=-\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}.[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div>Since\u00a0 [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y}[\/latex] or [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)}[\/latex] and we know the value for sine, we have<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\frac{1}{2}}=2.[\/latex][latex]\\\\[\/latex]<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">\n<div><\/div>\n<div style=\"text-align: left;\">Finally, since [latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y}[\/latex] or [latex]\\mathrm{cot}\\left(t\\right)=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}[\/latex] using the values for sine and cosine, we have<\/div>\n<div style=\"text-align: center;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}}{\\frac{1}{2}}=\\frac{-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\left(\\frac{2}{1}\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}.[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722103\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_05_03_01\">\n<div id=\"fs-id1165132947401\">\n<p>The point [latex]\\left(\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}\\right)[\/latex] is on the unit circle, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_02_003\">\u00a0Figure 4<\/a>.<\/p>\n<p id=\"fs-id1165134122782\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_05_03_002\" class=\"small\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<div class=\"small\">\n<div style=\"width: 411px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132418\/CNX_Precalc_Figure_05_03_003.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.\" width=\"401\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4:<\/strong>\u00a0 Graph of circle with angle of t inscribed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137471776\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137471776\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137471776\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137803942\">[latex]\\mathrm{sin}\\left(t\\right)=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{cos}\\left(t\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{tan}\\left(t\\right)=-1,\\text{ }\\mathrm{sec}\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{csc}\\left(t\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{cot}\\left(t\\right)=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766765\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_05_03_08\">\n<div id=\"fs-id1165137793763\">\n<p id=\"fs-id1165137793764\">Find the values of the six trigonometric functions of angle [latex]t[\/latex] based on <a class=\"autogenerated-content\" href=\"#Figure_05_03_010\">Figure 5<\/a><strong>. <\/strong><\/p>\n<div id=\"Figure_05_03_010\" 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\/3896\/2019\/03\/07132437\/CNX_Precalc_Figure_05_03_010.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"406\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5:<\/strong>\u00a0 Graph of circle with angle of t inscribed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135640611\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135640611\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135640611\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135640612\">[latex]\\begin{align*}\\mathrm{sin}\\left(t\\right)&=-1,\\\\\\mathrm{cos}\\left(t\\right)&=0,\\\\\\mathrm{tan}\\left(t\\right)&=\\text{Undefined}\\\\ \\mathrm{sec}\\left(t\\right)&=\\text{ Undefined,}\\\\\\mathrm{csc}\\left(t\\right)&=-1,\\\\\\mathrm{cot}\\left(t\\right)&=0\\end{align*}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_05_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165134094606\">\n<div id=\"fs-id1165137629063\">\n<h3>Example 2: Finding the Trigonometric Functions of an Angle<\/h3>\n<p id=\"fs-id1165137612022\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right)[\/latex] when [latex]t=\\frac{\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135508641\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135508641\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135508641\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137641007\">We have previously used the relationship of the sides of our special right triangles to demonstrate that [latex]\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex] and [latex]\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}.[\/latex] We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)}{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)}=\\frac{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=1[\/latex][latex]\\\\[\/latex]<\/p>\n<div><\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{sec}\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)} =\\frac{1}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}=\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{csc}\\left(\\frac{\\pi}{4}\\right)=\\frac{1}{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)}=\\frac{1}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}=\\frac{2\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{cot}\\left(\\frac{\\pi}{4}\\right)=\\frac{\\mathrm{cos}\\left(\\frac{\\pi}{4}\\right)}{\\mathrm{sin}\\left(\\frac{\\pi}{4}\\right)} =\\frac{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}{\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}}=1[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135496597\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_05_03_02\">\n<div id=\"fs-id1165135203430\">\n<p id=\"fs-id1165135203431\">Find [latex]\\mathrm{sin}\\left(t\\right),\\text{ }\\mathrm{cos}\\left(t\\right),\\text{ }\\mathrm{tan}\\left(t\\right),\\text{ }\\mathrm{sec}\\left(t\\right),\\text{ }\\mathrm{csc}\\left(t\\right),[\/latex] and [latex]\\mathrm{cot}\\left(t\\right)[\/latex] when [latex]t=\\frac{\\pi }{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132988446\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132988446\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132988446\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165132988447\">[latex]\\begin{align*}\\mathrm{sin}\\left(\\frac{\\pi }{3}\\right)&=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}\\\\ \\mathrm{cos}\\left(\\frac{\\pi }{3}\\right)&=\\frac{1}{2}\\\\ \\mathrm{tan}\\left(\\frac{\\pi }{3}\\right)&=\\sqrt[\\leftroot{1}\\uproot{2} ]{3}\\\\ \\mathrm{sec}\\left(\\frac{\\pi }{3}\\right)&=2\\\\ \\mathrm{csc}\\left(\\frac{\\pi }{3}\\right)&=\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}\\\\ \\mathrm{cot}\\left(\\frac{\\pi }{3}\\right)&=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}\\end{align*}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137598924\">Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting [latex]x[\/latex] equal to the cosine and [latex]y[\/latex] equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in <a class=\"autogenerated-content\" href=\"#Table_05_03_01\">Table 2<\/a>.<\/p>\n<table id=\"Table_05_03_01\" style=\"width: 1030px;\" summary=\"..\">\n<caption><strong>Table 2<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Angle<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\"><strong> [latex]0[\/latex] <\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\"><strong>[latex]\\frac{\\pi }{6},\\text{ or 30\u00b0}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\"><strong>[latex]\\frac{\\pi }{4},\\text{ or 45\u00b0}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\"><strong>[latex]\\frac{\\pi }{3},\\text{ or 60\u00b0}[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\"><strong>[latex]\\frac{\\pi }{2},\\text{ or 90\u00b0}[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cosine<\/strong><\/td>\n<td class=\"border\" style=\"width: 86.0781px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Sine<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">0<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Tangent<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">0<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">1<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">Undefined<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Secant<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">1<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">2<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">Undefined<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cosecant<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">Undefined<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">2<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{2}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 63.2812px;\"><strong>Cotangent<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 86.0781px;\">Undefined<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 219.781px;\">1<\/td>\n<td class=\"border\" style=\"text-align: center; width: 250.094px;\">[latex]\\frac{1}{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}\\text{ or }\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center; width: 84.6719px;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134284490\" class=\"bc-section section\">\n<div id=\"fs-id1321556\" class=\"bc-section section\">\n<h3>Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent<\/h3>\n<p id=\"fs-id1165137664668\">We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the <span class=\"no-emphasis\">reference angle<\/span> formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by <em>x\u00a0<\/em>and <em>y<\/em>-values in the original quadrant. <a class=\"autogenerated-content\" href=\"#Figure_05_03_004\">\u00a0Figure 6<\/a> shows which functions are positive in which quadrant.<\/p>\n<p id=\"fs-id1165137663666\">To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase \u201cA Smart Trig Class.\u201d Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is \u201c<strong>A<\/strong>,\u201d <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, \u201c<strong>S<\/strong>mart,\u201d only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, \u201c<strong>T<\/strong>rig,\u201d only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, \u201c<strong>C<\/strong>lass,\u201d only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.<\/p>\n<div id=\"Figure_05_03_004\" class=\"small\">\n<div style=\"width: 408px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132421\/CNX_Precalc_Figure_05_03_004.jpg\" alt=\"Graph of circle with each quadrant labeled. Under quadrant 1, labels fro sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant 2, labels for sin t and csc t. Under quadrant 3, labels for tan t and cot t. Under quadrant 4, labels for cos t, sec t.\" width=\"398\" height=\"297\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6:<\/strong> Quadrants with trig functions that are positive.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134468910\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137863545\"><strong>Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions. <\/strong><\/p>\n<ol id=\"fs-id1165135485968\" type=\"1\">\n<li>If the angle is not between [latex]0[\/latex] and [latex]2\\pi[\/latex] or [latex]0^{\\circ}[\/latex] and [latex]360^{\\circ}[\/latex], first add or subtract as many full revolutions as necessary in order to find a coterminal angle that is within these boundaries.<\/li>\n<li>Measure the angle formed by the terminal side of your angle and the horizontal axis. This is the reference angle.<\/li>\n<li>Evaluate the function at the reference angle.<\/li>\n<li>Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137735785\">\n<div id=\"fs-id1165135168424\">\n<h3>Example 3: Using Reference Angles to Find Trigonometric Functions<\/h3>\n<p id=\"fs-id1165133004515\">Use reference angles to find all six trigonometric functions of [latex]-\\frac{5\\pi }{6}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135704070\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135704070\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135704070\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135704072\">A coterminal angle within 0 and [latex]2\\pi[\/latex] will be [latex]\\frac{7\\pi}{6}.[\/latex] The angle between this angle\u2019s terminal side and the <em>x<\/em>-axis is [latex]\\frac{\\pi }{6},[\/latex] so that is the reference angle. Since [latex]-\\frac{5\\pi }{6}[\/latex] is in the third quadrant, where both [latex]x[\/latex] and [latex]y[\/latex] are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.<\/p>\n<div id=\"eip-id1165135700065\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{cos}\\left(-\\frac{5\\pi }{6}\\right)&=-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2},\\\\\\mathrm{sin}\\left(-\\frac{5\\pi }{6}\\right)&=-\\frac{1}{2},\\\\\\mathrm{tan}\\left(-\\frac{5\\pi }{6}\\right)&=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3} \\\\ \\mathrm{sec}\\left(-\\frac{5\\pi }{6}\\right)&=-\\frac{2\\text{ }\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3},\\\\\\mathrm{csc}\\left(-\\frac{5\\pi }{6}\\right)&=-2,\\\\\\mathrm{cot}\\left(-\\frac{5\\pi }{6}\\right)&=\\sqrt[\\leftroot{1}\\uproot{2} ]{3} \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627037\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_05_03_03\">\n<div id=\"fs-id1165137414778\">\n<p id=\"fs-id1165137476436\">Use reference angles to find all six trigonometric functions of [latex]-\\frac{7\\pi }{4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135196912\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135196912\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135196912\" class=\"hidden-answer\" style=\"display: none\"> A co-terminal angle is [latex]\\frac{\\pi}{4}[\/latex] and since this angle is in quadrant 1, it is also the reference angle.<\/p>\n<p id=\"fs-id1165135196914\">[latex]\\mathrm{sin}\\left(\\frac{-7\\pi }{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{cos}\\left(\\frac{-7\\pi }{4}\\right)=\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2},\\text{ }\\mathrm{tan}\\left(\\frac{-7\\pi }{4}\\right)=1,[\/latex]<\/p>\n<div><\/div>\n<p>[latex]\\mathrm{sec}\\left(\\frac{-7\\pi }{4}\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{csc}\\left(\\frac{-7\\pi }{4}\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{2},\\text{ }\\mathrm{cot}\\left(\\frac{-7\\pi }{4}\\right)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Using Even and Odd Trigonometric Functions<\/h3>\n<p id=\"fs-id1165137571788\">To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.<\/p>\n<p>Recall that:<\/p>\n<ul>\n<li id=\"fs-id1165134042164\">An <span class=\"no-emphasis\">even function<\/span> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right).[\/latex]<\/li>\n<li id=\"fs-id1165137443964\">An <span class=\"no-emphasis\">odd function<\/span> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\n<\/ul>\n<p><span style=\"font-size: 1rem; text-align: initial;\">We can test whether a trigonometric function is even or odd by drawing a <\/span><span class=\"no-emphasis\" style=\"font-size: 1rem; text-align: initial;\">unit circle<\/span><span style=\"font-size: 1rem; text-align: initial;\"> with a positive and a negative angle, as in <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Figure_05_03_007\">\u00a0Figure 7<\/a><span style=\"font-size: 1rem; text-align: initial;\">. The sine of the positive angle is [latex]y.[\/latex] The sine of the negative angle is \u2212<\/span><em style=\"font-size: 1rem; text-align: initial;\">y<\/em><span style=\"font-size: 1rem; text-align: initial;\">. The <\/span><span class=\"no-emphasis\" style=\"font-size: 1rem; text-align: initial;\">sine function<\/span><span style=\"font-size: 1rem; text-align: initial;\">, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in <\/span><a class=\"autogenerated-content\" style=\"font-size: 1rem; text-align: initial;\" href=\"#Table_05_03_02\">Table 3<\/a><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div id=\"Figure_05_03_007\" class=\"small\">\n<div style=\"width: 405px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132429\/CNX_Precalc_Figure_05_03_007.jpg\" alt=\"Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.\" width=\"395\" height=\"299\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7:<\/strong>\u00a0 Graph of circle with angle of t and -t inscribed.<\/p>\n<\/div>\n<\/div>\n<table id=\"Table_05_03_02\" class=\"aligncenter\" style=\"width: 769px;\" summary=\"..\">\n<caption><strong>Table 3<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{sin}\\text{ }\\left(t\\right)&=y\\\\ \\mathrm{sin}\\left(-t\\right)&=-y\\\\ \\mathrm{sin}\\left(-t\\right)&=\\mathrm{-sin}\\text{} \\left(t\\right)\\end{align*}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{cos}\\text{ }\\left(t\\right)&=x\\\\\\mathrm{cos}\\left(-t\\right)&=x\\\\ \\mathrm{cos}\\left(-t\\right)&=\\mathrm{cos}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{tan}\\left(t\\right)&=\\frac{y}{x}\\\\\\mathrm{tan}\\left(-t\\right)&=-\\frac{y}{x} \\\\ \\mathrm{tan}\\left(-t\\right)&=\\mathrm{-tan}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{csc}\\text{ }\\left(t\\right)&=\\frac{1}{y}\\\\\\mathrm{csc}\\left(-t\\right)&=\\frac{1}{-y}\\\\ \\mathrm{csc}\\left(-t\\right)&=\\mathrm{-csc}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{sec}\\text{ }\\left(t\\right)&=\\frac{1}{x}\\\\\\mathrm{sec}\\left(-t\\right)&=\\frac{1}{x}\\\\ \\mathrm{sec}\\left(-t\\right)&=\\mathrm{sec}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\n<td class=\"border\" style=\"width: 213.5px;\">[latex]\\begin{align*}\\mathrm{cot}\\text{ }\\left(t\\right)&=\\frac{x}{y}\\\\\\mathrm{cot}\\left(-t\\right)&=\\frac{x}{-y}\\\\ \\mathrm{cot}\\left(-t\\right)&=\\mathrm{-cot}\\text{ }\\left(t\\right)\\end{align*}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137824115\">\n<p id=\"fs-id1165137456021\">Therefore, we can see that cosine and secant are even and sine, tangent, cosecant, and cotangent are odd.<\/p>\n<div id=\"Example_05_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137559306\">\n<div id=\"fs-id1165137559308\">\n<h3>Example 4: Using Even and Odd Properties of Trigonometric Functions<\/h3>\n<p id=\"fs-id1165137736069\">If the secant of angle [latex]t[\/latex] is 2, what is the secant of [latex]-t?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134043531\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134043531\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134043531\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134043534\">Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle <em>t<\/em> is 2, the secant of [latex]-t[\/latex] is also 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137697003\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"ti_05_03_04\">\n<div id=\"fs-id1165135251458\">\n<p id=\"fs-id1165135251459\">If the cotangent of angle [latex]t[\/latex] is [latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3},[\/latex] what is the cotangent of [latex]-t?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137807232\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137807232\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137807232\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137807233\">[latex]-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404616\" class=\"bc-section section\">\n<div id=\"fs-id1165134267806\" class=\"bc-section section\">\n<h3>Recognizing and Using Fundamental Identities<\/h3>\n<p id=\"fs-id1165137655585\">We have now explored a number of\u00a0 definitions and properties of trigonometric functions and can use them to help us find values for other trigonometric function values for a specific angle.\u00a0 We can also use the definitions to help simplify trigonometric expressions.\u00a0 As you continue on to Calculus, you will see that it is oftentimes advantageous to work with the simplest expression possible.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137404616\" class=\"bc-section section\">\n<div id=\"fs-id1165134267806\" class=\"bc-section section\">\n<div id=\"Example_05_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135414324\">\n<div id=\"fs-id1165135414326\">\n<h3>Example 5: Using Identities to Simplify Trigonometric Expressions<\/h3>\n<p id=\"fs-id1165137433430\">Simplify [latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137456723\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137456723\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137456723\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137471602\">We can simplify this by rewriting both functions in terms of sine and cosine.<\/p>\n<div id=\"eip-id1165133223704\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}&=\\frac{\\frac{1}{\\mathrm{cos}\\left(t\\right)}}{\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}}&&\\text{ }\\\\\\text{ }&=\\frac{1}{\\mathrm{cos}\\left(t\\right)}\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}&&\\text{To divide the functions, multiply by the reciprocal}.\\\\\\text{ }&=\\frac{1}{\\mathrm{sin}\\left(t\\right)}&&\\text{Divide out the cosines}.\\\\\\text{ }&=\\mathrm{csc}\\left(t\\right)&&\\text{Simplify and use the identity. } \\end{align*}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165135496299\">By showing that [latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}[\/latex] can be simplified to [latex]\\mathrm{csc}\\left(t\\right),[\/latex] we have, in fact, established a new identity.<\/p>\n<div id=\"eip-id1165137404775\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}=\\mathrm{csc}\\left(t\\right).[\/latex]<\/div>\n<h3>Analysis<\/h3>\n<div id=\"fs-id1165135414324\">\n<div id=\"fs-id1165137456723\">\n<div>When simplifying trigonometric expressions, we need to consider the domain restrictions of the original expression before any algebra is done.\u00a0 The equivalent expression will only be valid on the original expression&#8217;s domain.\u00a0 The expression\u00a0[latex]\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}[\/latex] has restrictions which include those of the domain of the secant function, the domain of the tangent function and finally those where the tangent function is zero.<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div id=\"fs-id1165135414324\">\n<div id=\"fs-id1165137456723\">\n<div>\n<p>The domain of the tangent and secant function both exclude values where [latex]cos\\left(t\\right) = 0[\/latex] or where [latex]t= \\frac{\\pi}{2}[\/latex] or [latex]t= \\frac{3\\pi}{2}[\/latex] and any coterminal angles.\u00a0 This list of exceptions is often written as [latex]\\frac{\\pi}{2}+ \\pi k[\/latex] where [latex]k[\/latex] is an integer.\u00a0 Further, tangent is zero where [latex]sin\\left(t\\right)=0[\/latex] or where [latex]t=0[\/latex] or [latex]\\pi[\/latex] and any coterminal angles.\u00a0 These domain exceptions can be written as [latex]t = k \\pi[\/latex] where [latex]k[\/latex] is an integer.\u00a0 Therefore, our final equality<\/p>\n<div id=\"fs-id1165135414324\">\n<div id=\"fs-id1165137456723\">\n<div id=\"eip-id1165137404775\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\mathrm{sec}\\left(t\\right)}{\\mathrm{tan}\\left(t\\right)}=\\mathrm{csc}\\left(t\\right)\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>is only valid where [latex]sin\\left(t\\right)\\ne0[\/latex] and [latex]cos\\left(t\\right)\\ne0[\/latex] or where [latex]t\\ne\\frac{\\pi}{2}k[\/latex] for [latex]k[\/latex] an integer.<\/p>\n<\/div>\n<div class=\"unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137936650\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"ti_05_03_06\">\n<div id=\"fs-id1165137659799\">\n<p id=\"fs-id1165137659800\">Simplify [latex]\\left(\\mathrm{tan}\\left(t\\right)\\right)\\left(\\mathrm{cos}\\left(t\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134148506\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134148506\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134148506\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134148508\">[latex]\\mathrm{sin}\\left(t\\right)[\/latex]<\/p>\n<p>The domain of the tangent function excludes values where [latex]cos\\left(t\\right) = 0[\/latex] or where [latex]t= \\frac{\\pi}{2}[\/latex] or [latex]t= \\frac{3\\pi}{2}[\/latex] and any coterminal angles.\u00a0 This list of exceptions is often written as [latex]\\frac{\\pi}{2}+ \\pi k[\/latex] where [latex]k[\/latex] is an integer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732163\" class=\"bc-section section\">\n<h4>The Pythagorean Identity<\/h4>\n<p id=\"fs-id1165135203709\"><span style=\"font-size: 1rem; text-align: initial;\">You should recall that as a direct result of our definition of the sine and cosine functions in terms of the coordinates of points on the unit circle, we were able to create the Pythagorean Identity given below:\u00a0 \u00a0<\/span><\/p>\n<p style=\"text-align: center;\">[latex]{\\mathrm{cos}}^{2}\\left(\\theta\\right)+{\\mathrm{sin}}^{2}\\left(\\theta\\right)=1.[\/latex]<\/p>\n<p>This identity can often be used to find the sine or cosine function value if we know one of these values for a given angle.\u00a0 By using the other basic identities, we can then find the values of all of the trigonometric functions for a given angle.<\/p>\n<div id=\"Example_05_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137863436\">\n<div id=\"fs-id1165137863439\">\n<h3>Example 6: Using Identities to Relate Trigonometric Functions<\/h3>\n<p id=\"fs-id1165135516891\">If [latex]\\text{cos}\\left(t\\right)=\\frac{12}{13}[\/latex] and [latex]t[\/latex] is in quadrant IV, as shown in <a class=\"autogenerated-content\" href=\"#Figure_05_03_008\">\u00a0Figure 8<\/a>, find the values of the other five trigonometric functions.<\/p>\n<div id=\"Figure_05_03_008\" class=\"small\">\n<div style=\"width: 391px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132432\/CNX_Precalc_Figure_05_03_008.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (12\/13, y) is at intersection of terminal side of angle and edge of circle.\" width=\"381\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8:<\/strong>\u00a0 Graph of circle with angle of t inscribed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137589991\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137589991\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137589991\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135485151\">We can find the sine using the Pythagorean Identity, [latex]{\\mathrm{cos}}^{2}\\left(t\\right)+{\\mathrm{sin}}^{2}\\left(t\\right)=1,[\/latex] and the remaining functions by relating them to sine and cosine.<\/p>\n<div id=\"eip-id1165135191762\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}{\\left(\\frac{12}{13}\\right)}^{2}+\\mathrm{sin}^{2}\\left(t\\right)&=1 \\\\\\mathrm{sin}^{2}\\left(t\\right)&=1-{\\left(\\frac{12}{13}\\right)}^{2}\\\\ \\mathrm{sin}^{2}\\left(t\\right)&=1-\\frac{144}{169}\\\\ \\mathrm{sin}^{2}\\left(t\\right)&=\\frac{25}{169}\\\\\\mathrm{sin}\\left(t\\right)&=\\pm\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{25}{169}}\\\\\\mathrm{sin}\\left(t\\right)&=\\pm\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{25}}{\\sqrt[\\leftroot{1}\\uproot{2} ]{169}}\\\\ \\mathrm{sin}\\left(t\\right)&=\\pm\\frac{5}{13} \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135333148\">The sign of the sine depends on the <em>y<\/em>-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the <em>y<\/em>-values are negative, its sine is negative, [latex]-\\frac{5}{13}.[\/latex]<\/p>\n<p id=\"fs-id1165135403285\">The remaining functions can be calculated using identities relating them to sine and cosine.<\/p>\n<div id=\"fs-id1165137705474\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\mathrm{tan}\\left(t\\right)&=\\frac{\\mathrm{sin}\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}=\\frac{-\\frac{5}{13}}{\\frac{12}{13}}=-\\frac{5}{12} \\\\ \\mathrm{sec}\\left(t\\right)&=\\frac{1}{\\mathrm{cos}\\left(t\\right)}=\\frac{1}{\\frac{12}{13}}=\\frac{13}{12}\\\\ \\mathrm{csc}\\left(t\\right)&=\\frac{1}{\\mathrm{sin}\\left(t\\right)}=\\frac{1}{-\\frac{5}{13}}=-\\frac{13}{5}\\\\\\mathrm{cot}\\left(t\\right)&=\\frac{1}{\\mathrm{tan}\\left(t\\right)}=\\frac{1}{-\\frac{5}{12}}=-\\frac{12}{5}\\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137564622\" class=\"precalculus tryit\">\n<h3>Try it #7<\/h3>\n<div id=\"ti_05_03_07\">\n<div id=\"fs-id1165135570427\">\n<p id=\"fs-id1165137855086\">If [latex]\\mathrm{sec}\\left(t\\right)=-\\frac{17}{8}[\/latex] and [latex]0\\lt t<\\pi ,[\/latex] \u00a0find the values of the other five functions.\u00a0 Hint:\u00a0 First find [latex]\\mathrm{cos}\\left(t\\right)[\/latex] using a reciprocal relationship.<\/p>\n<\/div>\n<div id=\"fs-id1165137426588\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137426588\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137426588\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137426589\">[latex]\\mathrm{cos}\\left(t\\right)=-\\frac{8}{17},\\text{ }\\mathrm{sin}\\left(t\\right)=\\frac{15}{17},\\text{ }\\mathrm{tan}\\left(t\\right)=-\\frac{15}{8}[\/latex]<\/p>\n<div><\/div>\n<p>[latex]\\mathrm{csc}\\left(t\\right)=\\frac{17}{15},\\text{ }\\mathrm{cot}\\left(t\\right)=-\\frac{8}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137934400\" class=\"bc-section section\">\n<h3>Evaluating Trigonometric Functions with a Calculator<\/h3>\n<p id=\"fs-id1165135530385\">We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.<\/p>\n<p id=\"fs-id1165135555490\">Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.<\/p>\n<p id=\"fs-id1165137704595\">If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor [latex]\\frac{\\pi }{180}[\/latex] to convert the degrees to radians. To find the secant of [latex]30\u00b0,[\/latex] we could press<\/p>\n<div id=\"eip-887\" class=\"unnumbered\">(for a scientific calculator): [latex]\\frac{1}{30\u00d7\\frac{\\pi }{180}}\\text{COS}[\/latex]<\/div>\n<p id=\"fs-id1165135342785\">or<\/p>\n<div id=\"eip-148\" class=\"unnumbered\">(for a graphing calculator):\u00a0 [latex]\\frac{1}{\\mathrm{cos}\\left(\\frac{30\\pi }{180}\\right)}[\/latex]<\/div>\n<div id=\"fs-id1165135208418\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137758027\"><strong>Given an angle measure in radians, use a <span style=\"text-decoration: underline;\">scientific<\/span> calculator to find the cosecant. <\/strong><\/p>\n<ol id=\"fs-id1165137758031\" type=\"1\">\n<li>If the calculator has degree mode and radian mode, set it to radian mode.<\/li>\n<li>Enter: [latex]1\\text{ \/}[\/latex]<\/li>\n<li>Enter the value of the angle inside parentheses.<\/li>\n<li>Press the SIN key.<\/li>\n<li>Press the = key.<\/li>\n<\/ol>\n<p id=\"fs-id1165137597190\"><strong>Given an angle measure in radians, use a <span style=\"text-decoration: underline;\">graphing utility<\/span>\/calculator to find the cosecant. <\/strong><\/p>\n<ol id=\"fs-id1165137470238\" type=\"1\">\n<li>If the graphing utility has degree mode and radian mode, set it to radian mode.<\/li>\n<li>Enter: [latex]1\\text{ \/}[\/latex]<\/li>\n<li>Press the SIN key.<\/li>\n<li>Enter the value of the angle inside parentheses.<\/li>\n<li>Press the ENTER key.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_05_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137572560\">\n<div id=\"fs-id1165135380123\">\n<h3>Example 7:\u00a0 Evaluating the Cosecant Using Technology<\/h3>\n<p id=\"fs-id1165135380128\">Evaluate the cosecant of [latex]\\frac{5\\pi }{7}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135698644\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135698644\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135698644\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135698646\">For a scientific calculator, enter information as follows:<\/p>\n<div id=\"fs-id1165135698649\" class=\"unnumbered\">[latex]\\text{1 \/ ( 5 }\u00d7\\text{ }\\pi \\text{ \/ 7 ) SIN =}[\/latex]<\/div>\n<div id=\"eip-id1165133337484\" class=\"unnumbered\">[latex]\\mathrm{csc}\\left(\\frac{5\\pi }{7}\\right)\\approx 1.279[\/latex][latex]\\\\[\/latex]<\/div>\n<div>For a graphing calculator, enter the information as follows:<\/div>\n<div>[latex]1\/[\/latex] and now press the SIN key.\u00a0 Then enter [latex]\\frac{5\\pi}{7}[\/latex] followed by a closing parenthesis. Now press ENTER.<\/div>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div id=\"fs-id1165135698644\">\n<div><\/div>\n<div class=\"unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134216886\" class=\"precalculus tryit\">\n<h3>Try it #8<\/h3>\n<div id=\"ti_05_03_10\">\n<div id=\"fs-id1165134237307\">\n<p id=\"fs-id1165134237308\">Evaluate the cotangent of [latex]-\\frac{\\pi }{8}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137435028\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137435028\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137435028\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137435029\">[latex]\\approx -2.414[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593133\" class=\"precalculus media\">\n<div id=\"fs-id1165134284490\" class=\"bc-section section\">\n<div id=\"fs-id1321556\" class=\"bc-section section\">\n<h3>Analyzing the Graph of <em>y<\/em> = tan(<em>x)<\/em><\/h3>\n<p id=\"fs-id1530665\">We will begin with the graph of the <span class=\"no-emphasis\">tangent<\/span> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\n<div id=\"fs-id1614888\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\mathrm{tan}\\left(x\\right)=\\frac{\\mathrm{sin}\\left(x\\right)}{\\mathrm{cos}\\left(x\\right)}.[\/latex][latex]\\\\[\/latex]<\/div>\n<p>Remember that there are some values of [latex]x[\/latex] for which [latex]\\mathrm{cos}\\left(x\\right)=0.[\/latex] For example, [latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)=0[\/latex] and [latex]\\mathrm{cos}\\left(\\frac{3\\pi }{2}\\right)=0.[\/latex] At these values, the <span class=\"no-emphasis\">tangent function<\/span> is undefined, so the graph of [latex]y=\\mathrm{tan}\\left(x\\right)[\/latex] has discontinuities at [latex]x=\\frac{\\pi }{2}[\/latex] and [latex]\\frac{3\\pi }{2}.[\/latex] We will examine the function from a numerical point of view to see if there is evidence that there are vertical asymptotes at these points of discontinuity.<\/p>\n<p id=\"fs-id1647294\">We have already shown, using the ideas from the unit circle, that the tangent function is odd.<\/p>\n<p id=\"fs-id1460636\">We can further analyze the numerical behavior of the tangent function by looking at values for some of the special angles, as listed in <a class=\"autogenerated-content\" href=\"#Table_06_02_00\">Table 4<\/a>.<\/p>\n<table id=\"Table_06_02_00\" style=\"height: 94px;\" 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<caption><strong>Table 4<\/strong><\/caption>\n<tbody>\n<tr style=\"height: 54px;\">\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{2}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 54px; width: 70.5px; text-align: center;\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 40px;\">\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\"><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">undefined<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]-\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">\u20131<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70px; text-align: center;\">0<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">[latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{3}[\/latex]<\/td>\n<td class=\"border\" style=\"height: 40px; width: 70.5px; text-align: center;\">undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1279869\">These points will help us draw our graph, but we need to determine how the graph behaves where the function 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 1.5707,[\/latex] we will evaluate [latex]x[\/latex] at radian measures [latex]1.05 \\lt x \\lt 1.5707[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Table_06_02_01\">Table 5<\/a>.<\/p>\n<table id=\"Table_06_02_01\" style=\"width: 220px;\" 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<caption><strong>Table 5<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 38px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 30px; text-align: center;\">1.3<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.5<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.55<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.56<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1.57<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 38px; text-align: center;\"><strong>[latex]\\mathrm{tan} \\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 30px; text-align: center;\">3.6<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">14.1<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">48.1<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">92.6<\/td>\n<td class=\"border\" style=\"width: 38px; text-align: center;\">1255.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1658883\">As [latex]x[\/latex] approaches [latex]\\frac{\\pi }{2}[\/latex] from the left hand side, the outputs of the function get larger and larger or as [latex]x\\to{\\frac{\\pi}{2}}^{-},\\text{ }\\mathrm{tan}\\left(x\\right)\\to\\infty.[\/latex] This provides us with evidence that there is a vertical asymptote at\u00a0[latex]\\frac{\\pi }{2}.[\/latex]<\/p>\n<p>Because [latex]y=\\mathrm{tan}\\left(x\\right)[\/latex] is an odd function, we see the corresponding table of negative values in <a class=\"autogenerated-content\" href=\"#Table_06_02_02\">Table 6<\/a>.<\/p>\n<table id=\"Table_06_02_02\" style=\"height: 73px; width: 270px; width: 457px;\" 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<caption><strong>Table 6<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 188.494px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 33.0398px; text-align: center;\">\u22121.3<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.5<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.55<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u22121.56<\/td>\n<td class=\"border\" style=\"width: 18.4943px; text-align: center;\">-1.57<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 188.494px; text-align: center;\"><strong>[latex]\\mathrm{tan}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 33.0398px; text-align: center;\">\u22123.6<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221214.1<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221248.1<\/td>\n<td class=\"border\" style=\"width: 37.5852px; text-align: center;\">\u221292.6<\/td>\n<td class=\"border\" style=\"width: 18.4943px; text-align: center;\">-1255.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1318044\">We can see that, as [latex]x[\/latex] approaches [latex]-\\frac{\\pi }{2}[\/latex] from the right hand side, the outputs get more and more negative [latex]x\\to{\\frac{-\\pi}{2}}^{+},\\text{ }\\mathrm{tan}\\left(x\\right)\\to-\\infty.[\/latex]\u00a0 Again, this gives us evidence that these is a vertical asymptote at [latex]-\\frac{\\pi }{2}.[\/latex]<\/p>\n<p><a class=\"autogenerated-content\" href=\"#Figure_06_02_001\">Figure 9<\/a> represents the graph of [latex]y=\\mathrm{tan}\\left(x\\right).[\/latex] The tangent is positive from 0 to [latex]\\frac{\\pi }{2}[\/latex] and from [latex]\\pi[\/latex] to [latex]\\frac{3\\pi }{2},[\/latex] corresponding to quadrants I and III of the unit circle.<\/p>\n<p>We could create more points for other intervals and we will see that the tangent function repeats its behavior every [latex]\\pi[\/latex] units.\u00a0 We therefore conclude that the tangent function has a period of\u00a0[latex]\\pi[\/latex].<\/p>\n<div id=\"Figure_06_02_001\" class=\"small\">\n<div style=\"width: 472px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133144\/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=\"462\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9: Graph of the tangent function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<h3>Analyzing the Graphs of <em>y<\/em> = sec(<em>x)<\/em> and <em>y<\/em> = csc(<em>x)<\/em><\/h3>\n<p id=\"fs-id1405395\">The <span class=\"no-emphasis\">secant<\/span> was defined by the <span class=\"no-emphasis\">reciprocal identity<\/span> [latex]\\mathrm{sec}\\left(x\\right)=\\frac{1}{\\mathrm{cos}\\left(x\\right)}.[\/latex] Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi }{2},[\/latex] [latex]\\frac{3\\pi }{2},[\/latex] etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p id=\"fs-id1615608\">We can graph [latex]y=\\mathrm{sec}\\left(x\\right)[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_008\">Figure 10<\/a>. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <span class=\"no-emphasis\">secant function<\/span> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.<\/p>\n<p id=\"fs-id1657904\">The secant graph has vertical asymptotes at each value of [latex]x[\/latex] where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.<\/p>\n<p id=\"fs-id1457737\">Note that, because cosine is an even function, secant is also an even function. That is, [latex]\\mathrm{sec}\\left(-x\\right)=\\mathrm{sec}\\left(x\\right)[\/latex]<\/p>\n<div id=\"Figure_06_02_008\" class=\"small\">\n<div class=\"wp-caption-text\"><\/div>\n<div style=\"width: 393px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133203\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" width=\"383\" height=\"298\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10: Graph of the secant function, [latex]f\\left(x\\right)=\\mathrm{sec}\\left(x\\right)=\\frac{1}{\\mathrm{cos}\\left(x\\right).}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1700128\">\n<p id=\"fs-id1380717\">Similar to the secant, the <span class=\"no-emphasis\">cosecant<\/span> is defined by the reciprocal identity [latex]\\mathrm{csc}\\left(x\\right)=\\frac{1}{\\mathrm{sin}}\\left(x\\right).[\/latex] Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at [latex]0,[\/latex] [latex]\\pi,[\/latex] etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p id=\"fs-id1413697\">We can graph [latex]y=\\mathrm{csc}\\left(x\\right)[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">Figure 11<\/a>. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <span class=\"no-emphasis\">cosecant function<\/span> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\n<p id=\"fs-id1380921\">The cosecant graph has vertical asymptotes at each value of [latex]x[\/latex] where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\n<p id=\"fs-id1457402\">Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\mathrm{csc}\\left(-x\\right)=\\mathrm{-csc}\\left(x\\right).[\/latex]<\/p>\n<p id=\"eip-610\">The graph of cosecant, which is shown in <a class=\"autogenerated-content\" href=\"#Figure_06_02_009\">Figure 9<\/a>, is similar to the graph of secant.<\/p>\n<div id=\"Figure_06_02_009\" class=\"small\">\n<div style=\"width: 398px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133206\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" width=\"388\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11:<\/strong> The graph of the cosecant function, [latex]f\\left(x\\right)=\\mathrm{csc}\\left(x\\right)=\\frac{1}{\\mathrm{sin}\\left(x\\right).}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analyzing the Graph of <em>y<\/em> = cot(<em>x)<\/em><\/h3>\n<p id=\"fs-id1585113\">The last trigonometric function we need to explore is <span class=\"no-emphasis\">cotangent<\/span>. The cotangent is defined by the <span class=\"no-emphasis\">reciprocal identity<\/span> [latex]\\mathrm{cot}\\left(x\\right)=\\frac{1}{\\mathrm{tan}\\left(x\\right)}.[\/latex] Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at [latex]0,\\pi ,[\/latex] etc. Since the output of the tangent function is all real numbers, the output of the <span class=\"no-emphasis\">cotangent function<\/span> is also all real numbers.<\/p>\n<p id=\"fs-id1333454\">We can graph [latex]y=\\mathrm{cot}\\left(x\\right)[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See <a class=\"autogenerated-content\" href=\"#Figure_06_02_017\">Figure 12<\/a>. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\n<p id=\"fs-id1430801\">The cotangent graph has vertical asymptotes at each value of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)=0;[\/latex] we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\mathrm{cot}\\left(x\\right)[\/latex] has vertical asymptotes at all values of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)=0,[\/latex] and [latex]\\mathrm{cot}\\left(x\\right)=0[\/latex] at all values of [latex]x[\/latex] where [latex]\\mathrm{tan}\\left(x\\right)[\/latex] has its vertical asymptotes.<\/p>\n<div id=\"Figure_06_02_017\" class=\"small\">\n<div class=\"wp-caption-text\"><\/div>\n<div style=\"width: 342px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133237\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"332\" height=\"299\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12:<\/strong> The cotangent function.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771142\">\n<h3>Period of a Function<\/h3>\n<p>As we have previously discussed, a function that repeats its values in regular intervals is known as a <span class=\"no-emphasis\">periodic function<\/span>.\u00a0 For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or [latex]2\\pi ,[\/latex] will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.<\/p>\n<p id=\"fs-id1165135593361\">Remember, the period [latex]P[\/latex] of a repeating function [latex]f[\/latex] is the number representing the interval such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex] for any value of [latex]x.[\/latex]<\/p>\n<p id=\"fs-id1165135333706\">The period of the cosine, sine, secant, and cosecant functions is [latex]2\\pi .[\/latex]<\/p>\n<p id=\"fs-id1165137893417\">The period of the tangent and cotangent functions is [latex]\\pi .[\/latex]<span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">\u00a0<\/span><\/p>\n<p>Other functions can also be periodic. For example, the lengths of months repeat every four years. If [latex]x[\/latex] represents the length time, measured in years, and [latex]f\\left(x\\right)[\/latex] represents the number of days in February, then [latex]f\\left(x+4\\right)=f\\left(x\\right).[\/latex] This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A <strong>period<\/strong> is the shortest interval over which a function completes one full cycle\u2014in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135191384\">Access these online resources for additional instruction and practice with other trigonometric functions.<\/p>\n<ul id=\"fs-id1165137473589\">\n<li><a href=\"http:\/\/openstax.org\/l\/trigfuncval\">Determining Trig Function Values<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/moretrigfun\">More Examples of Determining Trig Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/pythagiden\">Pythagorean Identities<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/trigcalc\">Trig Functions on a Calculator<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137938685\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134112952\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 116.531px;\">Tangent function<\/td>\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{tan}\\left(t\\right)=\\frac{\\mathrm{sin}t\\left(t\\right)}{\\mathrm{cos}\\left(t\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 116.531px;\">Secant function<\/td>\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{\\mathrm{cos}\\left(t\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 116.531px;\">Cosecant function<\/td>\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{\\mathrm{sin}\\left(t\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 116.531px;\">Cotangent function<\/td>\n<td class=\"border\" style=\"width: 719.469px;\">[latex]\\mathrm{cot}\\left(t\\right)=\\frac{1}{\\mathrm{tan}\\left(t\\right)}=\\frac{\\mathrm{cos}\\left(t\\right)}{\\mathrm{sin}\\left(t\\right)}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137832791\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul>\n<li>The tangent of an angle is the ratio of the <em>y<\/em>-value to the <em>x<\/em>-value of the corresponding point on the unit circle.<\/li>\n<li>Secant, cotangent, and cosecant are all reciprocals of other functions. The secant function is the reciprocal of the cosine function, the cotangent function is the reciprocal of the tangent function, and the cosecant function is the reciprocal of the sine function.<\/li>\n<li>The six trigonometric functions can be found from a point on the unit circle.<\/li>\n<li>Trigonometric functions can also be found from an angle.<\/li>\n<li>Trigonometric functions of angles outside the first quadrant can be determined using reference angles.<\/li>\n<li>A function is said to be even if [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex] and odd if [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1165134211396\">\n<li>Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.<\/li>\n<li>Even and odd properties can be used to evaluate trigonometric functions.<\/li>\n<li>The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.<\/li>\n<li>Identities can be used to evaluate trigonometric functions.<\/li>\n<li>Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities.<\/li>\n<li>The trigonometric functions repeat at regular intervals.<\/li>\n<li>The period [latex]P[\/latex] of a repeating function [latex]f[\/latex] is the smallest interval such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex] for any value of [latex]x.[\/latex]<\/li>\n<li>The values of trigonometric functions of special angles can be found by mathematical analysis.<\/li>\n<li>To evaluate trigonometric functions of other angles, we can use a calculator or computer software.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137640892\">\n<dt>cosecant<\/dt>\n<dd id=\"fs-id1165137640897\">the reciprocal of the sine function: on the unit circle, [latex]\\mathrm{csc}\\left(t\\right)=\\frac{1}{y},y\\ne 0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137529833\">\n<dt>cotangent<\/dt>\n<dd id=\"fs-id1165137529838\">the reciprocal of the tangent function: on the unit circle, [latex]\\mathrm{cot}\\left(t\\right)=\\frac{x}{y},y\\ne 0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137410476\">\n<dt>identities<\/dt>\n<dd id=\"fs-id1165137410481\">statements that are true for all values of the input on which they are defined<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135195519\">\n<dt>period<\/dt>\n<dd id=\"fs-id1165135195524\">the smallest interval [latex]P[\/latex] of a repeating function [latex]f[\/latex] such that [latex]f\\left(x+P\\right)=f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135591029\">\n<dt>secant<\/dt>\n<dd id=\"fs-id1165135591035\">the reciprocal of the cosine function: on the unit circle, [latex]\\mathrm{sec}\\left(t\\right)=\\frac{1}{x},x\\ne 0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137834349\">\n<dt>tangent<\/dt>\n<dd id=\"fs-id1165137834355\">the quotient of the sine and cosine: on the unit circle, [latex]\\mathrm{tan}\\left(t\\right)=\\frac{y}{x},x\\ne 0[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2990\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Other Trigonometric Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:3xp253uc@9\/The-Other-Trigonometric-Functions\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:3xp253uc@9\/The-Other-Trigonometric-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":158108,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Other Trigonometric Functions\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:3xp253uc@9\/The-Other-Trigonometric-Functions\",\"project\":\"Essential Precalcus, Part 2\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2990","chapter","type-chapter","status-publish","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2990","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158108"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2990\/revisions"}],"predecessor-version":[{"id":3280,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2990\/revisions\/3280"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2990\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2990"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2990"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2990"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}