{"id":180,"date":"2025-02-13T22:44:42","date_gmt":"2025-02-13T22:44:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/unit-circle-sine-and-cosine-functions\/"},"modified":"2026-03-24T06:38:04","modified_gmt":"2026-03-24T06:38:04","slug":"unit-circle-sine-and-cosine-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/unit-circle-sine-and-cosine-functions\/","title":{"raw":"Sine and Cosine Functions: Learn It 1","rendered":"Sine and Cosine 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 function values for the sine and cosine of the special angles.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use reference angles to evaluate trigonometric functions.<\/li>\r\n \t<li style=\"font-weight: 400;\">Evaluate sine and cosine values using a calculator.<\/li>\r\n<\/ul>\r\n<\/section>To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1. The angle (in radians) that [latex]t[\/latex] intercepts forms an arc of length [latex]s[\/latex]. Using the formula [latex]s=rt[\/latex], and knowing that [latex]r=1[\/latex], we see that for a <strong>unit circle<\/strong>, [latex]s=t[\/latex].\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <em>x- <\/em>and <em>y-<\/em>axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.<\/section>For any angle [latex]t[\/latex], we can label the intersection of the terminal side and the unit circle as by its coordinates, [latex]\\left(x,y\\right)[\/latex]. The coordinates [latex]x[\/latex] and [latex]y[\/latex] will be the outputs of the trigonometric functions [latex]f\\left(t\\right)=\\cos t[\/latex] and [latex]f\\left(t\\right)=\\sin t[\/latex], respectively. This means [latex]x=\\cos t[\/latex] and [latex]y=\\sin t[\/latex].\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\/3675\/2018\/09\/27003528\/CNX_Precalc_Figure_05_02_0022.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and an arc created by the angle with length s. The terminal side of the angle intersects the circle at the point (x,y).\" width=\"487\" height=\"385\" \/> Unit circle where the central angle is [latex]t[\/latex] radians[\/caption]<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>unit circle<\/h3>\r\nA <strong>unit circle<\/strong> has a center at [latex]\\left(0,0\\right)[\/latex] and radius [latex]1[\/latex] . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle [latex]1[\/latex].\r\n\r\n&nbsp;\r\n\r\nLet [latex]\\left(x,y\\right)[\/latex] be the endpoint on the unit circle of an arc of arc length [latex]s[\/latex]. The [latex]\\left(x,y\\right)[\/latex] coordinates of this point can be described as functions of the angle.\r\n\r\n<\/section>\r\n<h2>Defining Sine and Cosine Functions<\/h2>\r\nNow that we have our unit circle labeled, we can learn how the [latex]\\left(x,y\\right)[\/latex] coordinates relate to the <strong>arc length<\/strong> and <strong>angle<\/strong>. The <strong>sine function<\/strong> relates a real number [latex]t[\/latex] to the <em>y<\/em>-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex]. In Figure 2, the sine is equal to [latex]y[\/latex]. Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the <em>y<\/em>-coordinate of the corresponding point on the unit circle.\r\n\r\nThe <strong>cosine function<\/strong> of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex]. In Figure 3, the cosine is equal to [latex]x[\/latex].<span id=\"fs-id1165137735775\">\r\n<\/span>\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\/3675\/2018\/09\/27003530\/CNX_Precalc_Figure_05_02_0032.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=\"487\" height=\"226\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n&nbsp;\r\n\r\nBecause it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\\sin t[\/latex] is the same as [latex]\\sin \\left(t\\right)[\/latex] and [latex]\\cos t[\/latex]\u00a0is the same as [latex]\\cos \\left(t\\right)[\/latex]. Likewise, [latex]{\\cos }^{2}t[\/latex] is a commonly used shorthand notation for [latex]{\\left(\\cos \\left(t\\right)\\right)}^{2}[\/latex]. Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>sine and cosine functions<\/h3>\r\nIf [latex]t[\/latex] is a real number and a point [latex]\\left(x,y\\right)[\/latex] on the unit circle corresponds to an angle of [latex]t[\/latex], then\r\n<div style=\"text-align: center;\">[latex]\\cos t=x[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex]\\sin t=y[\/latex]<\/div>\r\n<\/section>\r\n<div><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a point <em>P<\/em> [latex]\\left(x,y\\right)[\/latex] on the unit circle corresponding to an angle of [latex]t[\/latex], find the sine and cosine.<\/strong>\r\n<ol>\r\n \t<li>The sine of [latex]t[\/latex] is equal to the <em>y<\/em>-coordinate of point [latex]P:\\sin t=y[\/latex].<\/li>\r\n \t<li>The cosine of [latex]t[\/latex] is equal to the <em>x<\/em>-coordinate of point [latex]P: \\text{cos}t=x[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Point [latex]P[\/latex] is a point on the unit circle corresponding to an angle of [latex]t[\/latex]. Find [latex]\\cos \\left(t\\right)[\/latex] and [latex]\\text{sin}\\left(t\\right)[\/latex].<span id=\"fs-id1165137723705\">\r\n<\/span><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003532\/CNX_Precalc_Figure_05_02_0042.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1\/2, square root of 3 over 2).\" width=\"487\" height=\"385\" \/>[reveal-answer q=\"288317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288317\"]We know that [latex]\\cos t[\/latex] is the <em>x<\/em>-coordinate of the corresponding point on the unit circle and [latex]\\sin t[\/latex] is the <em>y<\/em>-coordinate of the corresponding point on the unit circle. So:\r\n<p style=\"text-align: center;\">[latex]\\begin{align} x&amp;=\\cos t=\\frac{1}{2} \\\\ y&amp;=\\sin t=\\frac{\\sqrt{3}}{2} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">A certain angle [latex]t[\/latex] corresponds to a point on the unit circle at [latex]\\left(-\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2}\\right)[\/latex] as shown in Figure 5. Find [latex]\\cos t[\/latex]\u00a0and [latex]\\sin t[\/latex].<span id=\"fs-id1165137434263\">\r\n<\/span>\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\/3675\/2018\/09\/27003535\/CNX_Precalc_Figure_05_02_0052.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).\" width=\"487\" height=\"383\" \/> <b>Figure 5<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"110315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"110315\"]\r\n\r\n[latex]\\cos \\left(t\\right)=-\\frac{\\sqrt{2}}{2},\\sin \\left(t\\right)=\\frac{\\sqrt{2}}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Finding Sines and Cosines of Angles on an Axis<\/h2>\r\nFor quadrantral angles, the corresponding point on the unit circle falls on the <em>x- <\/em>or <em>y<\/em>-axis. In that case, we can easily calculate cosine and sine from the values of [latex]x[\/latex] and [latex]y[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\cos \\left(90^\\circ \\right)[\/latex] and [latex]\\text{sin}\\left(90^\\circ \\right)[\/latex].[reveal-answer q=\"736685\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"736685\"]Moving [latex]90^\\circ [\/latex] counterclockwise around the unit circle from the positive <em>x<\/em>-axis brings us to the top of the circle, where the [latex]\\left(x,y\\right)[\/latex] coordinates are (0, 1).<span id=\"fs-id1165135641569\">\r\n<\/span><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003537\/CNX_Precalc_Figure_05_02_0062.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).\" width=\"487\" height=\"383\" \/>Using our definitions of cosine and sine,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\cos t=\\cos \\left(90^\\circ \\right)=0\\\\ y&amp;=\\sin t=\\sin \\left(90^\\circ \\right)=1\\end{align}[\/latex]<\/p>\r\nThe cosine of [latex]90^\\circ [\/latex] is [latex]0[\/latex]; the sine of [latex]90^\\circ [\/latex] is [latex]1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Find cosine and sine of the angle [latex]\\pi [\/latex].[reveal-answer q=\"479772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"479772\"][latex]\\cos \\left(\\pi \\right)=-1[\/latex], [latex]\\sin \\left(\\pi \\right)=0[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find function values for the sine and cosine of the special angles.<\/li>\n<li style=\"font-weight: 400;\">Use reference angles to evaluate trigonometric functions.<\/li>\n<li style=\"font-weight: 400;\">Evaluate sine and cosine values using a calculator.<\/li>\n<\/ul>\n<\/section>\n<p>To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1. The angle (in radians) that [latex]t[\/latex] intercepts forms an arc of length [latex]s[\/latex]. Using the formula [latex]s=rt[\/latex], and knowing that [latex]r=1[\/latex], we see that for a <strong>unit circle<\/strong>, [latex]s=t[\/latex].<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <em>x- <\/em>and <em>y-<\/em>axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.<\/section>\n<p>For any angle [latex]t[\/latex], we can label the intersection of the terminal side and the unit circle as by its coordinates, [latex]\\left(x,y\\right)[\/latex]. The coordinates [latex]x[\/latex] and [latex]y[\/latex] will be the outputs of the trigonometric functions [latex]f\\left(t\\right)=\\cos t[\/latex] and [latex]f\\left(t\\right)=\\sin t[\/latex], respectively. This means [latex]x=\\cos t[\/latex] and [latex]y=\\sin t[\/latex].<\/p>\n<figure style=\"width: 487px\" 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\/27003528\/CNX_Precalc_Figure_05_02_0022.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and an arc created by the angle with length s. The terminal side of the angle intersects the circle at the point (x,y).\" width=\"487\" height=\"385\" \/><figcaption class=\"wp-caption-text\">Unit circle where the central angle is [latex]t[\/latex] radians<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>unit circle<\/h3>\n<p>A <strong>unit circle<\/strong> has a center at [latex]\\left(0,0\\right)[\/latex] and radius [latex]1[\/latex] . In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle [latex]1[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Let [latex]\\left(x,y\\right)[\/latex] be the endpoint on the unit circle of an arc of arc length [latex]s[\/latex]. The [latex]\\left(x,y\\right)[\/latex] coordinates of this point can be described as functions of the angle.<\/p>\n<\/section>\n<h2>Defining Sine and Cosine Functions<\/h2>\n<p>Now that we have our unit circle labeled, we can learn how the [latex]\\left(x,y\\right)[\/latex] coordinates relate to the <strong>arc length<\/strong> and <strong>angle<\/strong>. The <strong>sine function<\/strong> relates a real number [latex]t[\/latex] to the <em>y<\/em>-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle [latex]t[\/latex] equals the <em>y<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex]. In Figure 2, the sine is equal to [latex]y[\/latex]. Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the <em>y<\/em>-coordinate of the corresponding point on the unit circle.<\/p>\n<p>The <strong>cosine function<\/strong> of an angle [latex]t[\/latex] equals the <em>x<\/em>-value of the endpoint on the unit circle of an arc of length [latex]t[\/latex]. In Figure 3, the cosine is equal to [latex]x[\/latex].<span id=\"fs-id1165137735775\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 487px\" 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\/27003530\/CNX_Precalc_Figure_05_02_0032.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=\"487\" height=\"226\" \/><figcaption class=\"wp-caption-text\"><b>Figure 3<\/b><\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\\sin t[\/latex] is the same as [latex]\\sin \\left(t\\right)[\/latex] and [latex]\\cos t[\/latex]\u00a0is the same as [latex]\\cos \\left(t\\right)[\/latex]. Likewise, [latex]{\\cos }^{2}t[\/latex] is a commonly used shorthand notation for [latex]{\\left(\\cos \\left(t\\right)\\right)}^{2}[\/latex]. Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>sine and cosine functions<\/h3>\n<p>If [latex]t[\/latex] is a real number and a point [latex]\\left(x,y\\right)[\/latex] on the unit circle corresponds to an angle of [latex]t[\/latex], then<\/p>\n<div style=\"text-align: center;\">[latex]\\cos t=x[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]\\sin t=y[\/latex]<\/div>\n<\/section>\n<div>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a point <em>P<\/em> [latex]\\left(x,y\\right)[\/latex] on the unit circle corresponding to an angle of [latex]t[\/latex], find the sine and cosine.<\/strong><\/p>\n<ol>\n<li>The sine of [latex]t[\/latex] is equal to the <em>y<\/em>-coordinate of point [latex]P:\\sin t=y[\/latex].<\/li>\n<li>The cosine of [latex]t[\/latex] is equal to the <em>x<\/em>-coordinate of point [latex]P: \\text{cos}t=x[\/latex].<\/li>\n<\/ol>\n<\/section>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Point [latex]P[\/latex] is a point on the unit circle corresponding to an angle of [latex]t[\/latex]. Find [latex]\\cos \\left(t\\right)[\/latex] and [latex]\\text{sin}\\left(t\\right)[\/latex].<span id=\"fs-id1165137723705\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003532\/CNX_Precalc_Figure_05_02_0042.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1\/2, square root of 3 over 2).\" width=\"487\" height=\"385\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q288317\">Show Solution<\/button><\/p>\n<div id=\"q288317\" class=\"hidden-answer\" style=\"display: none\">We know that [latex]\\cos t[\/latex] is the <em>x<\/em>-coordinate of the corresponding point on the unit circle and [latex]\\sin t[\/latex] is the <em>y<\/em>-coordinate of the corresponding point on the unit circle. So:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} x&=\\cos t=\\frac{1}{2} \\\\ y&=\\sin t=\\frac{\\sqrt{3}}{2} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">A certain angle [latex]t[\/latex] corresponds to a point on the unit circle at [latex]\\left(-\\frac{\\sqrt{2}}{2},\\frac{\\sqrt{2}}{2}\\right)[\/latex] as shown in Figure 5. Find [latex]\\cos t[\/latex]\u00a0and [latex]\\sin t[\/latex].<span id=\"fs-id1165137434263\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 487px\" 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\/27003535\/CNX_Precalc_Figure_05_02_0052.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).\" width=\"487\" height=\"383\" \/><figcaption class=\"wp-caption-text\"><b>Figure 5<\/b><\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q110315\">Show Solution<\/button><\/p>\n<div id=\"q110315\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\cos \\left(t\\right)=-\\frac{\\sqrt{2}}{2},\\sin \\left(t\\right)=\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding Sines and Cosines of Angles on an Axis<\/h2>\n<p>For quadrantral angles, the corresponding point on the unit circle falls on the <em>x- <\/em>or <em>y<\/em>-axis. In that case, we can easily calculate cosine and sine from the values of [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\cos \\left(90^\\circ \\right)[\/latex] and [latex]\\text{sin}\\left(90^\\circ \\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q736685\">Show Solution<\/button><\/p>\n<div id=\"q736685\" class=\"hidden-answer\" style=\"display: none\">Moving [latex]90^\\circ[\/latex] counterclockwise around the unit circle from the positive <em>x<\/em>-axis brings us to the top of the circle, where the [latex]\\left(x,y\\right)[\/latex] coordinates are (0, 1).<span id=\"fs-id1165135641569\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003537\/CNX_Precalc_Figure_05_02_0062.jpg\" alt=\"Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).\" width=\"487\" height=\"383\" \/>Using our definitions of cosine and sine,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\cos t=\\cos \\left(90^\\circ \\right)=0\\\\ y&=\\sin t=\\sin \\left(90^\\circ \\right)=1\\end{align}[\/latex]<\/p>\n<p>The cosine of [latex]90^\\circ[\/latex] is [latex]0[\/latex]; the sine of [latex]90^\\circ[\/latex] is [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Find cosine and sine of the angle [latex]\\pi[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q479772\">Show Solution<\/button><\/p>\n<div id=\"q479772\" class=\"hidden-answer\" style=\"display: none\">[latex]\\cos \\left(\\pi \\right)=-1[\/latex], [latex]\\sin \\left(\\pi \\right)=0[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Trigonometric Functions Using Angles\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vaG4O6d48mo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Unit Circle Image\",\"author\":\"CK-12\",\"organization\":\"\",\"url\":\"http:\/\/www.ck12.org\/trigonometry\/Unit-Circle\/lesson\/Trigonometric-Ratios-on-the-Unit-Circle\/\",\"project\":\"\",\"license\":\"cc-by-nc\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""},{"type":"copyrighted_video","description":"Introduction to Trigonometric Functions Using Angles","author":"Mathispower4u","organization":"","url":"https:\/\/youtu.be\/vaG4O6d48mo","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"cc","description":"Unit Circle Image","author":"CK-12","organization":"","url":"http:\/\/www.ck12.org\/trigonometry\/Unit-Circle\/lesson\/Trigonometric-Ratios-on-the-Unit-Circle\/","project":"","license":"cc-by-nc","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/180"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":16,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/180\/revisions"}],"predecessor-version":[{"id":5974,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/180\/revisions\/5974"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/180\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=180"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=180"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=180"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}