{"id":181,"date":"2025-02-13T22:44:43","date_gmt":"2025-02-13T22:44:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions\/"},"modified":"2025-10-09T19:16:31","modified_gmt":"2025-10-09T19:16:31","slug":"the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-other-trigonometric-functions\/","title":{"raw":"The Other Trigonometric Functions: Learn It 1","rendered":"The Other Trigonometric Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\r\n \t<li style=\"font-weight: 400;\">Use properties of even and odd trigonometric functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Recognize and use fundamental identities.<\/li>\r\n \t<li style=\"font-weight: 400;\">Evaluate trigonometric functions with a calculator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent<\/h2>\r\nTo define the remaining functions, we will once again draw a unit circle with a point [latex]\\left(x,y\\right)[\/latex] corresponding to an angle of [latex]t[\/latex],. As with the sine and cosine, we can use the [latex]\\left(x,y\\right)[\/latex] coordinates to find the other functions.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003658\/CNX_Precalc_Figure_05_03_0012.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (x, y) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"198\" \/>\r\n\r\nThe first function we will define is the tangent. The <strong>tangent<\/strong> 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. The tangent of angle [latex]t[\/latex] is equal to [latex]\\frac{y}{x},x\\ne 0[\/latex]. Because the <em>y<\/em>-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{\\sin t}{\\cos t},\\cos t\\ne 0[\/latex]. The tangent function is abbreviated as [latex]\\tan[\/latex]. The remaining three functions can all be expressed as reciprocals of functions we have already defined.\r\n<ul>\r\n \t<li>The <strong>secant<\/strong> function is the reciprocal of the cosine function. The secant of angle [latex]t[\/latex] is equal to [latex]\\frac{1}{\\cos t}=\\frac{1}{x},x\\ne 0[\/latex]. The secant function is abbreviated as [latex]\\sec[\/latex].<\/li>\r\n \t<li>The <strong>cotangent<\/strong> function is the reciprocal of the tangent function. The cotangent of angle [latex]t[\/latex] is equal to [latex]\\frac{\\cos t}{\\sin t}=\\frac{x}{y},y\\ne 0[\/latex]. The cotangent function is abbreviated as [latex]\\cot[\/latex].<\/li>\r\n \t<li>The <strong>cosecant<\/strong> function is the reciprocal of the sine function. The cosecant of angle [latex]t[\/latex] is equal to [latex]\\frac{1}{\\sin t}=\\frac{1}{y},y\\ne 0[\/latex]. The cosecant function is abbreviated as [latex]\\csc[\/latex].<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Tangent, Secant, Cosecant, and Cotangent<\/h3>\r\nIf [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\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\tan t=\\frac{y}{x},x\\ne 0\\\\ \\sec t=\\frac{1}{x},x\\ne 0\\\\ \\csc t=\\frac{1}{y},y\\ne 0\\\\ \\cot t=\\frac{x}{y},y\\ne 0\\end{gathered}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The point [latex]\\left(-\\frac{\\sqrt{3}}{2},\\frac{1}{2}\\right)[\/latex] is on the unit circle. Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex].<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003700\/CNX_Precalc_Figure_05_03_0022.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=\"487\" height=\"216\" \/>[reveal-answer q=\"714608\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"714608\"]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:\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin t=y=\\frac{1}{2}\\\\ \\cos t=x=-\\frac{\\sqrt{3}}{2}\\\\ \\tan t=\\frac{y}{x}=\\frac{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}=\\frac{1}{2}\\left(-\\frac{2}{\\sqrt{3}}\\right)=-\\frac{1}{\\sqrt{3}}=-\\frac{\\sqrt{3}}{3}\\\\ \\sec t=\\frac{1}{x}=\\frac{1}{\\frac{-\\frac{\\sqrt{3}}{2}}{}}=-\\frac{2}{\\sqrt{3}}=-\\frac{2\\sqrt{3}}{3}\\\\ \\csc t=\\frac{1}{y}=\\frac{1}{\\frac{1}{2}}=2\\\\ \\cot t=\\frac{x}{y}=\\frac{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}=-\\frac{\\sqrt{3}}{2}\\left(\\frac{2}{1}\\right)=-\\sqrt{3}\\end{gathered}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nThe point [latex]\\left(\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2}\\right)[\/latex] is on the unit circle. Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003703\/CNX_Precalc_Figure_05_03_0032.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=\"487\" height=\"347\" \/>\r\n\r\n[reveal-answer q=\"264592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"264592\"]\r\n\r\n[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2},\\tan t=-1,\\sec t=\\sqrt{2},\\csc t=-\\sqrt{2},\\cot t=-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex] when [latex]t=\\frac{\\pi }{6}[\/latex].[reveal-answer q=\"419551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"419551\"]We have previously used the properties of equilateral triangles to demonstrate that [latex]\\sin \\frac{\\pi }{6}=\\frac{1}{2}[\/latex] and [latex]\\cos \\frac{\\pi }{6}=\\frac{\\sqrt{3}}{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.\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} \\tan \\frac{\\pi }{6}=\\frac{\\sin\\frac{\\pi }{6}}{\\cos \\frac{\\pi }{6}} =\\frac{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}=\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}\\end{gathered}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sec \\frac{\\pi }{6}=\\frac{1}{\\cos \\frac{\\pi }{6}} =\\frac{1}{\\frac{\\sqrt{3}}{2}}=\\frac{2}{\\sqrt{3}}=\\frac{2\\sqrt{3}}{3}\\end{gathered}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\csc \\frac{\\pi }{6}=\\frac{1}{\\sin \\frac{\\pi }{6}}=\\frac{1}{\\frac{1}{2}}=2\\end{gathered}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem;\">[latex]\\begin{gathered}\\cot \\frac{\\pi }{6}=\\frac{\\cos \\frac{\\pi }{6}}{\\sin \\frac{\\pi }{6}} =\\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}=\\sqrt{3} \\end{gathered}[\/latex]<\/span><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex] when [latex]t=\\frac{\\pi }{3}[\/latex].[reveal-answer q=\"151548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151548\"][latex]\\begin{align}&amp;\\sin \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{2}\\\\ &amp;\\cos \\frac{\\pi }{3}=\\frac{1}{2}\\\\ &amp;\\tan \\frac{\\pi }{3}=\\sqrt{3}\\\\ &amp;\\sec \\frac{\\pi }{3}=2\\\\ &amp;\\csc \\frac{\\pi }{3}=\\frac{2\\sqrt{3}}{3}\\\\ &amp;\\cot \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{3}\\end{align}[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]173354[\/ohm_question]<\/section>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 the table below.\r\n<table id=\"Table_05_03_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Angle<\/strong><\/td>\r\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\r\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosine<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Sine<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Tangent<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Secant<\/strong><\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>Undefined<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cosecant<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>2<\/td>\r\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Cotangent<\/strong><\/td>\r\n<td>Undefined<\/td>\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section id=\"fs-id1165137832791\" class=\"key-concepts\">\r\n<div><\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find exact values of the other trigonometric functions secant, cosecant, tangent, and cotangent<\/li>\n<li style=\"font-weight: 400;\">Use properties of even and odd trigonometric functions.<\/li>\n<li style=\"font-weight: 400;\">Recognize and use fundamental identities.<\/li>\n<li style=\"font-weight: 400;\">Evaluate trigonometric functions with a calculator.<\/li>\n<\/ul>\n<\/section>\n<h2>Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent<\/h2>\n<p>To define the remaining functions, we will once again draw a unit circle with a point [latex]\\left(x,y\\right)[\/latex] corresponding to an angle of [latex]t[\/latex],. As with the sine and cosine, we can use the [latex]\\left(x,y\\right)[\/latex] coordinates to find the other functions.<\/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\/27003658\/CNX_Precalc_Figure_05_03_0012.jpg\" alt=\"Graph of circle with angle of t inscribed. Point of (x, y) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"198\" \/><\/p>\n<p>The first function we will define is the tangent. The <strong>tangent<\/strong> 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. The tangent of angle [latex]t[\/latex] is equal to [latex]\\frac{y}{x},x\\ne 0[\/latex]. Because the <em>y<\/em>-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{\\sin t}{\\cos t},\\cos t\\ne 0[\/latex]. The tangent function is abbreviated as [latex]\\tan[\/latex]. The remaining three functions can all be expressed as reciprocals of functions we have already defined.<\/p>\n<ul>\n<li>The <strong>secant<\/strong> function is the reciprocal of the cosine function. The secant of angle [latex]t[\/latex] is equal to [latex]\\frac{1}{\\cos t}=\\frac{1}{x},x\\ne 0[\/latex]. The secant function is abbreviated as [latex]\\sec[\/latex].<\/li>\n<li>The <strong>cotangent<\/strong> function is the reciprocal of the tangent function. The cotangent of angle [latex]t[\/latex] is equal to [latex]\\frac{\\cos t}{\\sin t}=\\frac{x}{y},y\\ne 0[\/latex]. The cotangent function is abbreviated as [latex]\\cot[\/latex].<\/li>\n<li>The <strong>cosecant<\/strong> function is the reciprocal of the sine function. The cosecant of angle [latex]t[\/latex] is equal to [latex]\\frac{1}{\\sin t}=\\frac{1}{y},y\\ne 0[\/latex]. The cosecant function is abbreviated as [latex]\\csc[\/latex].<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Tangent, Secant, Cosecant, and Cotangent<\/h3>\n<p>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<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\tan t=\\frac{y}{x},x\\ne 0\\\\ \\sec t=\\frac{1}{x},x\\ne 0\\\\ \\csc t=\\frac{1}{y},y\\ne 0\\\\ \\cot t=\\frac{x}{y},y\\ne 0\\end{gathered}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The point [latex]\\left(-\\frac{\\sqrt{3}}{2},\\frac{1}{2}\\right)[\/latex] is on the unit circle. Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex].<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003700\/CNX_Precalc_Figure_05_03_0022.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=\"487\" height=\"216\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q714608\">Show Solution<\/button><\/p>\n<div id=\"q714608\" class=\"hidden-answer\" style=\"display: none\">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:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sin t=y=\\frac{1}{2}\\\\ \\cos t=x=-\\frac{\\sqrt{3}}{2}\\\\ \\tan t=\\frac{y}{x}=\\frac{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}=\\frac{1}{2}\\left(-\\frac{2}{\\sqrt{3}}\\right)=-\\frac{1}{\\sqrt{3}}=-\\frac{\\sqrt{3}}{3}\\\\ \\sec t=\\frac{1}{x}=\\frac{1}{\\frac{-\\frac{\\sqrt{3}}{2}}{}}=-\\frac{2}{\\sqrt{3}}=-\\frac{2\\sqrt{3}}{3}\\\\ \\csc t=\\frac{1}{y}=\\frac{1}{\\frac{1}{2}}=2\\\\ \\cot t=\\frac{x}{y}=\\frac{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}=-\\frac{\\sqrt{3}}{2}\\left(\\frac{2}{1}\\right)=-\\sqrt{3}\\end{gathered}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>The point [latex]\\left(\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2}\\right)[\/latex] is on the unit circle. Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex].<\/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\/27003703\/CNX_Precalc_Figure_05_03_0032.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=\"487\" height=\"347\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q264592\">Show Solution<\/button><\/p>\n<div id=\"q264592\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\sin t=-\\frac{\\sqrt{2}}{2},\\cos t=\\frac{\\sqrt{2}}{2},\\tan t=-1,\\sec t=\\sqrt{2},\\csc t=-\\sqrt{2},\\cot t=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex] when [latex]t=\\frac{\\pi }{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q419551\">Show Solution<\/button><\/p>\n<div id=\"q419551\" class=\"hidden-answer\" style=\"display: none\">We have previously used the properties of equilateral triangles to demonstrate that [latex]\\sin \\frac{\\pi }{6}=\\frac{1}{2}[\/latex] and [latex]\\cos \\frac{\\pi }{6}=\\frac{\\sqrt{3}}{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]\\begin{gathered} \\tan \\frac{\\pi }{6}=\\frac{\\sin\\frac{\\pi }{6}}{\\cos \\frac{\\pi }{6}} =\\frac{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}=\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}\\end{gathered}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\sec \\frac{\\pi }{6}=\\frac{1}{\\cos \\frac{\\pi }{6}} =\\frac{1}{\\frac{\\sqrt{3}}{2}}=\\frac{2}{\\sqrt{3}}=\\frac{2\\sqrt{3}}{3}\\end{gathered}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}\\csc \\frac{\\pi }{6}=\\frac{1}{\\sin \\frac{\\pi }{6}}=\\frac{1}{\\frac{1}{2}}=2\\end{gathered}[\/latex]<\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem;\">[latex]\\begin{gathered}\\cot \\frac{\\pi }{6}=\\frac{\\cos \\frac{\\pi }{6}}{\\sin \\frac{\\pi }{6}} =\\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}=\\sqrt{3} \\end{gathered}[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Find [latex]\\sin t,\\cos t,\\tan t,\\sec t,\\csc t[\/latex], and [latex]\\cot t[\/latex] when [latex]t=\\frac{\\pi }{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q151548\">Show Solution<\/button><\/p>\n<div id=\"q151548\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align}&\\sin \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{2}\\\\ &\\cos \\frac{\\pi }{3}=\\frac{1}{2}\\\\ &\\tan \\frac{\\pi }{3}=\\sqrt{3}\\\\ &\\sec \\frac{\\pi }{3}=2\\\\ &\\csc \\frac{\\pi }{3}=\\frac{2\\sqrt{3}}{3}\\\\ &\\cot \\frac{\\pi }{3}=\\frac{\\sqrt{3}}{3}\\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm173354\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173354&theme=lumen&iframe_resize_id=ohm173354&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>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 the table below.<\/p>\n<table id=\"Table_05_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Angle<\/strong><\/td>\n<td><strong> [latex]0[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\n<td><strong> [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Cosine<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td><strong>Sine<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Tangent<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Secant<\/strong><\/td>\n<td>1<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>2<\/td>\n<td>Undefined<\/td>\n<\/tr>\n<tr>\n<td><strong>Cosecant<\/strong><\/td>\n<td>Undefined<\/td>\n<td>2<\/td>\n<td>[latex]\\sqrt{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td><strong>Cotangent<\/strong><\/td>\n<td>Undefined<\/td>\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section 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