{"id":3767,"date":"2021-05-12T20:24:06","date_gmt":"2021-05-12T20:24:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/1c\/"},"modified":"2021-05-12T20:24:06","modified_gmt":"2021-05-12T20:24:06","slug":"1c","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/1c\/","title":{"raw":"Skills Review for Trigonometric Functions","rendered":"Skills Review for Trigonometric Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Identify reference angles for angles measured in both radians and degrees<\/li>\n \t<li>Evaluate trigonometric functions using the unit circle<\/li>\n \t<li>Use properties of even and odd trigonometric functions<\/li>\n<\/ul>\n<\/div>\nIn the Trigonometric Functions section, you will learn how to evaluate trigonometric functions at various angle measures and also graph trigonometric functions. Understanding how to find a reference angle of a given angle is an important skill needed to evaluate trigonometric functions and is reviewed here. Even-odd properties are also reviewed here, which will both help with evaluating trigonometric functions and graphing them.\n<h2>Find Reference Angles<\/h2>\nYou will learn that it is easiest to evaluate trigonometric functions when an angle is in the first quadrant. When the original angle is given in quadrant two, three, or four, a reference angle should be found.\n\nAn angle\u2019s <strong>reference angle<\/strong> is the acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. A reference angle is always an angle between [latex]0[\/latex] and [latex]90^\\circ [\/latex], or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see in the figure below, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003604\/CNX_Precalc_Figure_05_01_0195.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"975\" height=\"331\"> <b>A visual of the corresponding reference angles for each of the quadrants.<\/b>[\/caption]\n\n<div class=\"textbox\">\n<h3>How To: Given an angle between [latex]0[\/latex] and [latex]2\\pi [\/latex], find its reference angle.<\/h3>\n<ol>\n \t<li>An angle in the first quadrant is its own reference angle.<\/li>\n \t<li>For an angle in the second or third quadrant, the reference angle is [latex]|\\pi -t|[\/latex] or [latex]|180^\\circ \\mathrm{-t}|[\/latex].<\/li>\n \t<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^\\circ \\mathrm{-t}[\/latex].<\/li>\n \t<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi [\/latex], add or subtract [latex]2\\pi [\/latex] as many times as needed to find an equivalent angle between [latex]0[\/latex] and [latex]2\\pi [\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Reference Angle<\/h3>\nFind the reference angle of [latex]225^\\circ [\/latex] as shown in below.\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003606\/CNX_Precalc_Figure_05_02_0162.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"487\" height=\"383\">\n\n[reveal-answer q=\"770468\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"770468\"]\n\nBecause [latex]225^\\circ [\/latex] is in the third quadrant, the reference angle is\n<p style=\"text-align: center;\">[latex]|\\left(180^\\circ -225^\\circ \\right)|=|-45^\\circ |=45^\\circ [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nFind the reference angle of [latex]\\frac{5\\pi }{3}[\/latex].\n\n[reveal-answer q=\"227547\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"227547\"]\n\n[latex]\\frac{\\pi }{3}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nWe can evaluate trigonometric functions of angles outside the first quadrant using reference angles. The quadrant of the original angle determines whether the answer is positive or negative. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase \"A Smart Trig Class.\" 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 \"<strong>A<\/strong>,\" <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, \"<strong>S<\/strong>mart,\" only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, \"<strong>T<\/strong>rig,\" only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, \"<strong>C<\/strong>lass,\" only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003705\/CNX_Precalc_Figure_05_03_0042.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=\"487\" height=\"363\"> <b>An illustration of which trigonometric functions are positive in each of the quadrants.<\/b>[\/caption]\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\nThe unit circle tells us the value of cosine and sine at any of the given angle measures seen below. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. You will learn in Section 1.3 that all trigonometric functions can be written in terms of sine and cosine. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant of the unit circle.\n\nRemember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function's value at a given angle.\n\n<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\"><\/a>\n<div class=\"textbox\">\n<h3>How To: Given the angle of a point on The Unit circle, find the Value of Cosine (Or Sine) using quadrant one.<\/h3>\n<ol>\n \t<li>Find the reference angle using the appropriate reference angle formula from the first portion of this review section.<\/li>\n \t<li>Find the value of cosine (or sine) at the reference angle by looking at quadrant one of the unit circle.<\/li>\n \t<li>Determine the appropriate sign of your found value for cosine (or sine) based on the quadrant of the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Unit Circle to Find the Value of cosine<\/h3>\nUse quadrant one of the unit circle to find the value of cosine at an angle of [latex]\\frac{7\\pi }{6}[\/latex].\n\n[reveal-answer q=\"865133\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"865133\"]\n\nWe know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.\n\nFirst, let\u2019s find the reference angle. The reference angle is:\n<p style=\"text-align: center;\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex]<\/p>\nNext, we find the value of cosine at the reference angle which is represented by the first coordinate of the ordered pair at&nbsp;[latex]\\frac{\\pi }{6}[\/latex].\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\nBecause our original angle is in the third quadrant, where cosine is always negative, we have:\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nUse quadrant one of the unit circle to find the value of sine at an angle of [latex]\\frac{5\\pi }{3}[\/latex].\n\n[reveal-answer q=\"913342\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"913342\"]\n\n[latex]-\\frac{\\sqrt{3}}{2}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Determine Even and Odd Trigonometric Functions<\/h2>\nTo 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.\n\nConsider the function [latex]f\\left(x\\right)={x}^{2}[\/latex], shown below. The graph of the function is symmetrical about the <em>y<\/em>-axis. All along the curve, any two points with opposite <em>x<\/em>-values have the same function value. This matches the result of calculation: [latex]{\\left(4\\right)}^{2}={\\left(-4\\right)}^{2}[\/latex], [latex]{\\left(-5\\right)}^{2}={\\left(5\\right)}^{2}[\/latex],&nbsp;and so on. So [latex]f\\left(x\\right)={x}^{2}[\/latex] is an <strong>even function<\/strong>, a function such that two inputs that are opposites have the same output. That means [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].\n\n<span id=\"fs-id1165137817732\"> <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003707\/CNX_Precalc_Figure_05_03_0052.jpg\" alt=\"Graph of parabola with points (-2, 4) and (2, 4) labeled.\"><\/span>\n<p style=\"text-align: center;\">The function [latex]f\\left(x\\right)={x}^{2}[\/latex]&nbsp;is an even function.<\/p>\nNow consider the function [latex]f\\left(x\\right)={x}^{3}[\/latex], shown below. The graph is not symmetrical about the <em>y<\/em>-axis. All along the graph, any two points with opposite <em>x<\/em>-values also have opposite <em>y<\/em>-values. So [latex]f\\left(x\\right)={x}^{3}[\/latex] is an <strong>odd function<\/strong>, one such that two inputs that are opposites have outputs that are also opposites. That means [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].\n\n<span id=\"fs-id1165135545756\"> <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003710\/CNX_Precalc_Figure_05_03_0062.jpg\" alt=\"Graph of function with labels for points (-1, -1) and (1, 1).\"><\/span>\n<p style=\"text-align: center;\">The function [latex]f\\left(x\\right)={x}^{3}[\/latex]&nbsp;is an odd function.<\/p>\n\n<div class=\"textbox\">\n<h3>A General Note: Even and Odd Trigonometric Functions<\/h3>\nAn <strong>even function<\/strong> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].\n\nAn <strong>odd function<\/strong> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].\n\nCosine and secant are even:\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\cos \\left(-t\\right)=\\cos t \\\\ \\sec \\left(-t\\right)=\\sec t \\end{gathered}[\/latex]<\/p>\nSine, tangent, cosecant, and cotangent are odd:\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(-t\\right)=-\\sin t \\\\ \\tan \\left(-t\\right)=-\\tan t \\\\ \\csc \\left(-t\\right)=-\\csc t \\\\ \\cot \\left(-t\\right)=-\\cot t \\end{gathered}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Even and Odd Properties of Trigonometric Functions<\/h3>\nIf [latex]\\sec t=2[\/latex], what is the [latex]\\sec (-t)[\/latex]?\n\n[reveal-answer q=\"5363\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"5363\"]\n\nSecant 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.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nIf [latex]\\cot t=\\sqrt{3}[\/latex], what is [latex]\\cot (-t)[\/latex]?\n\n[reveal-answer q=\"840134\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"840134\"]\n\n[latex]-\\sqrt{3}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n[embed]https:\/\/youtu.be\/YbU8Sq0quWE[\/embed]\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify reference angles for angles measured in both radians and degrees<\/li>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<li>Use properties of even and odd trigonometric functions<\/li>\n<\/ul>\n<\/div>\n<p>In the Trigonometric Functions section, you will learn how to evaluate trigonometric functions at various angle measures and also graph trigonometric functions. Understanding how to find a reference angle of a given angle is an important skill needed to evaluate trigonometric functions and is reviewed here. Even-odd properties are also reviewed here, which will both help with evaluating trigonometric functions and graphing them.<\/p>\n<h2>Find Reference Angles<\/h2>\n<p>You will learn that it is easiest to evaluate trigonometric functions when an angle is in the first quadrant. When the original angle is given in quadrant two, three, or four, a reference angle should be found.<\/p>\n<p>An angle\u2019s <strong>reference angle<\/strong> is the acute angle, [latex]t[\/latex], formed by the terminal side of the angle [latex]t[\/latex] and the horizontal axis. A reference angle is always an angle between [latex]0[\/latex] and [latex]90^\\circ[\/latex], or [latex]0[\/latex] and [latex]\\frac{\\pi }{2}[\/latex] radians. As we can see in the figure below, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003604\/CNX_Precalc_Figure_05_01_0195.jpg\" alt=\"Four side by side graphs. First graph shows an angle of t in quadrant 1 in it's normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.\" width=\"975\" height=\"331\" \/><\/p>\n<p class=\"wp-caption-text\"><b>A visual of the corresponding reference angles for each of the quadrants.<\/b><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an angle between [latex]0[\/latex] and [latex]2\\pi[\/latex], find its reference angle.<\/h3>\n<ol>\n<li>An angle in the first quadrant is its own reference angle.<\/li>\n<li>For an angle in the second or third quadrant, the reference angle is [latex]|\\pi -t|[\/latex] or [latex]|180^\\circ \\mathrm{-t}|[\/latex].<\/li>\n<li>For an angle in the fourth quadrant, the reference angle is [latex]2\\pi -t[\/latex] or [latex]360^\\circ \\mathrm{-t}[\/latex].<\/li>\n<li>If an angle is less than [latex]0[\/latex] or greater than [latex]2\\pi[\/latex], add or subtract [latex]2\\pi[\/latex] as many times as needed to find an equivalent angle between [latex]0[\/latex] and [latex]2\\pi[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding a Reference Angle<\/h3>\n<p>Find the reference angle of [latex]225^\\circ[\/latex] as shown in below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003606\/CNX_Precalc_Figure_05_02_0162.jpg\" alt=\"Graph of circle with 225 degree angle inscribed.\" width=\"487\" height=\"383\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q770468\">Show Solution<\/span><\/p>\n<div id=\"q770468\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because [latex]225^\\circ[\/latex] is in the third quadrant, the reference angle is<\/p>\n<p style=\"text-align: center;\">[latex]|\\left(180^\\circ -225^\\circ \\right)|=|-45^\\circ |=45^\\circ[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the reference angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q227547\">Show Solution<\/span><\/p>\n<div id=\"q227547\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can evaluate trigonometric functions of angles outside the first quadrant using reference angles. The quadrant of the original angle determines whether the answer is positive or negative. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase &#8220;A Smart Trig Class.&#8221; 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 &#8220;<strong>A<\/strong>,&#8221; <strong><u>a<\/u><\/strong>ll of the six trigonometric functions are positive. In quadrant II, &#8220;<strong>S<\/strong>mart,&#8221; only <strong><u>s<\/u><\/strong>ine and its reciprocal function, cosecant, are positive. In quadrant III, &#8220;<strong>T<\/strong>rig,&#8221; only <strong><u>t<\/u><\/strong>angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, &#8220;<strong>C<\/strong>lass,&#8221; only <strong><u>c<\/u><\/strong>osine and its reciprocal function, secant, are positive.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003705\/CNX_Precalc_Figure_05_03_0042.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=\"487\" height=\"363\" \/><\/p>\n<p class=\"wp-caption-text\"><b>An illustration of which trigonometric functions are positive in each of the quadrants.<\/b><\/p>\n<\/div>\n<h2>Evaluate Trigonometric Functions Using the Unit Circle<\/h2>\n<p>The unit circle tells us the value of cosine and sine at any of the given angle measures seen below. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. You will learn in Section 1.3 that all trigonometric functions can be written in terms of sine and cosine. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Take time to learn the [latex]\\left(x,y\\right)[\/latex] coordinates of all of the major angles in the first quadrant of the unit circle.<\/p>\n<p>Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function&#8217;s value at a given angle.<\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a><\/p>\n<div class=\"textbox\">\n<h3>How To: Given the angle of a point on The Unit circle, find the Value of Cosine (Or Sine) using quadrant one.<\/h3>\n<ol>\n<li>Find the reference angle using the appropriate reference angle formula from the first portion of this review section.<\/li>\n<li>Find the value of cosine (or sine) at the reference angle by looking at quadrant one of the unit circle.<\/li>\n<li>Determine the appropriate sign of your found value for cosine (or sine) based on the quadrant of the original angle.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Unit Circle to Find the Value of cosine<\/h3>\n<p>Use quadrant one of the unit circle to find the value of cosine at an angle of [latex]\\frac{7\\pi }{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q865133\">Show Solution<\/span><\/p>\n<div id=\"q865133\" class=\"hidden-answer\" style=\"display: none\">\n<p>We know that the angle [latex]\\frac{7\\pi }{6}[\/latex] is in the third quadrant.<\/p>\n<p>First, let\u2019s find the reference angle. The reference angle is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{7\\pi }{6}-\\pi =\\frac{\\pi }{6}[\/latex]<\/p>\n<p>Next, we find the value of cosine at the reference angle which is represented by the first coordinate of the ordered pair at&nbsp;[latex]\\frac{\\pi }{6}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p>Because our original angle is in the third quadrant, where cosine is always negative, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\cos \\left(\\frac{7\\pi }{6}\\right)=-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Use quadrant one of the unit circle to find the value of sine at an angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q913342\">Show Solution<\/span><\/p>\n<div id=\"q913342\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determine Even and Odd Trigonometric Functions<\/h2>\n<p>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>Consider the function [latex]f\\left(x\\right)={x}^{2}[\/latex], shown below. The graph of the function is symmetrical about the <em>y<\/em>-axis. All along the curve, any two points with opposite <em>x<\/em>-values have the same function value. This matches the result of calculation: [latex]{\\left(4\\right)}^{2}={\\left(-4\\right)}^{2}[\/latex], [latex]{\\left(-5\\right)}^{2}={\\left(5\\right)}^{2}[\/latex],&nbsp;and so on. So [latex]f\\left(x\\right)={x}^{2}[\/latex] is an <strong>even function<\/strong>, a function such that two inputs that are opposites have the same output. That means [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].<\/p>\n<p><span id=\"fs-id1165137817732\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003707\/CNX_Precalc_Figure_05_03_0052.jpg\" alt=\"Graph of parabola with points (-2, 4) and (2, 4) labeled.\" \/><\/span><\/p>\n<p style=\"text-align: center;\">The function [latex]f\\left(x\\right)={x}^{2}[\/latex]&nbsp;is an even function.<\/p>\n<p>Now consider the function [latex]f\\left(x\\right)={x}^{3}[\/latex], shown below. The graph is not symmetrical about the <em>y<\/em>-axis. All along the graph, any two points with opposite <em>x<\/em>-values also have opposite <em>y<\/em>-values. So [latex]f\\left(x\\right)={x}^{3}[\/latex] is an <strong>odd function<\/strong>, one such that two inputs that are opposites have outputs that are also opposites. That means [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].<\/p>\n<p><span id=\"fs-id1165135545756\"> <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003710\/CNX_Precalc_Figure_05_03_0062.jpg\" alt=\"Graph of function with labels for points (-1, -1) and (1, 1).\" \/><\/span><\/p>\n<p style=\"text-align: center;\">The function [latex]f\\left(x\\right)={x}^{3}[\/latex]&nbsp;is an odd function.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Even and Odd Trigonometric Functions<\/h3>\n<p>An <strong>even function<\/strong> is one in which [latex]f\\left(-x\\right)=f\\left(x\\right)[\/latex].<\/p>\n<p>An <strong>odd function<\/strong> is one in which [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex].<\/p>\n<p>Cosine and secant are even:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\cos \\left(-t\\right)=\\cos t \\\\ \\sec \\left(-t\\right)=\\sec t \\end{gathered}[\/latex]<\/p>\n<p>Sine, tangent, cosecant, and cotangent are odd:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin \\left(-t\\right)=-\\sin t \\\\ \\tan \\left(-t\\right)=-\\tan t \\\\ \\csc \\left(-t\\right)=-\\csc t \\\\ \\cot \\left(-t\\right)=-\\cot t \\end{gathered}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Even and Odd Properties of Trigonometric Functions<\/h3>\n<p>If [latex]\\sec t=2[\/latex], what is the [latex]\\sec (-t)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5363\">Show Solution<\/span><\/p>\n<div id=\"q5363\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>If [latex]\\cot t=\\sqrt{3}[\/latex], what is [latex]\\cot (-t)[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840134\">Show Solution<\/span><\/p>\n<div id=\"q840134\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Even and Odd Trigonometric Identities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/YbU8Sq0quWE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-3767\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <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":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"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-3767","chapter","type-chapter","status-publish","hentry"],"part":3764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3767","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3767\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/3764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3767\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3767"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3767"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3767"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3767"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}