{"id":182,"date":"2025-02-13T22:44:44","date_gmt":"2025-02-13T22:44:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/right-triangle-trigonometry\/"},"modified":"2025-10-17T21:07:28","modified_gmt":"2025-10-17T21:07:28","slug":"right-triangle-trigonometry","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/right-triangle-trigonometry\/","title":{"raw":"Right Triangle Trigonometry: Learn It 1","rendered":"Right Triangle Trigonometry: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use right triangles to evaluate trigonometric functions.<\/li>\r\n \t<li>Use cofunctions of complementary angles.<\/li>\r\n \t<li>Use the de\ufb01nitions of trigonometric functions of any angle.<\/li>\r\n \t<li>Use right triangle trigonometry to solve applied problems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using Right Triangles to Evaluate Trigonometric Functions<\/h2>\r\nIn earlier sections, we used a unit circle to define the <strong>trigonometric functions<\/strong>. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of [latex]t[\/latex] is its value at [latex]t[\/latex] radians. First, we need to create our right triangle. If we drop a vertical line segment from the point [latex](x,y) [\/latex] to the <em>x<\/em>-axis, we have a right triangle whose vertical side has length [latex]y[\/latex] and whose horizontal side has length [latex]x[\/latex]. We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.<span id=\"fs-id1165137602828\">\r\n<\/span>\r\n\r\n&nbsp;\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003730\/CNX_Precalc_Figure_05_04_0012.jpg\" alt=\"Graph of quarter circle with radius of 1 and angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"208\" \/>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<div style=\"text-align: left;\">[latex]\\cos t=x[\/latex] and [latex]\\sin t=y[\/latex]<\/div>\r\n<\/section>These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using [latex]\\left(x,y\\right)[\/latex] coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of [latex]x[\/latex], we will call the side between the given angle and the right angle the <strong>adjacent side<\/strong> to angle [latex]t[\/latex]. (Adjacent means \"next to.\") Instead of [latex]y[\/latex], we will call the side most distant from the given angle the <strong>opposite side<\/strong> from angle [latex]t[\/latex]. And instead of [latex]1[\/latex], we will call the side of a right triangle opposite the right angle the <strong>hypotenuse<\/strong>.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>labeling right triangle sides<\/h3>\r\n<span id=\"fs-id1165137465030\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003732\/CNX_Precalc_Figure_05_04_0022.jpg\" alt=\"A right triangle with hypotenuse, opposite, and adjacent sides labeled.\" \/><\/span>\r\n<p style=\"text-align: center;\">The sides of a right triangle in relation to angle [latex]t[\/latex].<\/p>\r\n\r\n<\/section>\r\n<h2>Understanding Right Triangle Relationships<\/h2>\r\nGiven a right triangle with an acute angle of [latex]t[\/latex],\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;\\sin \\left(t\\right)=\\frac{\\text{opposite}}{\\text{hypotenuse}} \\\\ &amp;\\cos \\left(t\\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}} \\\\ &amp;\\tan \\left(t\\right)=\\frac{\\text{opposite}}{\\text{adjacent}} \\end{align}[\/latex]<\/div>\r\nA common mnemonic for remembering these relationships is SOH-CAH-TOA, formed from the first letters of \"<u>S<\/u>ine is <u>o<\/u>pposite over <u>h<\/u>ypotenuse, <u>C<\/u>osine is <u>a<\/u>djacent over <u>h<\/u>ypotenuse, <u>T<\/u>angent is <u>o<\/u>pposite over <u>a<\/u>djacent.\"\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.<\/strong>\r\n<ol>\r\n \t<li>Find the sine as the ratio of the opposite side to the hypotenuse.<\/li>\r\n \t<li>Find the cosine as the ratio of the adjacent side to the hypotenuse.<\/li>\r\n \t<li>Find the tangent is the ratio of the opposite side to the adjacent side.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the triangle, find the value of [latex]\\cos \\alpha[\/latex].<span id=\"fs-id1165137414609\">\r\n<\/span><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003734\/CNX_Precalc_Figure_05_04_0032.jpg\" alt=\"A right triangle with sid lengths of 8, 15, and 17. Angle alpha also labeled.\" width=\"487\" height=\"188\" \/>[reveal-answer q=\"880775\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880775\"]The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha \\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}} =\\frac{15}{17} \\end{align}[\/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\nGiven the triangle, find the value of [latex]\\text{sin}t[\/latex].<span id=\"fs-id1165135191134\">\r\n<\/span>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003736\/CNX_Precalc_Figure_05_04_0042.jpg\" alt=\"A right triangle with sides of 7, 24, and 25. Also labeled is angle t.\" width=\"487\" height=\"180\" \/>\r\n\r\n[reveal-answer q=\"261765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261765\"]\r\n\r\n[latex]\\frac{7}{25}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use right triangles to evaluate trigonometric functions.<\/li>\n<li>Use cofunctions of complementary angles.<\/li>\n<li>Use the de\ufb01nitions of trigonometric functions of any angle.<\/li>\n<li>Use right triangle trigonometry to solve applied problems.<\/li>\n<\/ul>\n<\/section>\n<h2>Using Right Triangles to Evaluate Trigonometric Functions<\/h2>\n<p>In earlier sections, we used a unit circle to define the <strong>trigonometric functions<\/strong>. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of [latex]t[\/latex] is its value at [latex]t[\/latex] radians. First, we need to create our right triangle. If we drop a vertical line segment from the point [latex](x,y)[\/latex] to the <em>x<\/em>-axis, we have a right triangle whose vertical side has length [latex]y[\/latex] and whose horizontal side has length [latex]x[\/latex]. We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.<span id=\"fs-id1165137602828\"><br \/>\n<\/span><\/p>\n<p>&nbsp;<\/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\/27003730\/CNX_Precalc_Figure_05_04_0012.jpg\" alt=\"Graph of quarter circle with radius of 1 and angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.\" width=\"487\" height=\"208\" \/><\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<div style=\"text-align: left;\">[latex]\\cos t=x[\/latex] and [latex]\\sin t=y[\/latex]<\/div>\n<\/section>\n<p>These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using [latex]\\left(x,y\\right)[\/latex] coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of [latex]x[\/latex], we will call the side between the given angle and the right angle the <strong>adjacent side<\/strong> to angle [latex]t[\/latex]. (Adjacent means &#8220;next to.&#8221;) Instead of [latex]y[\/latex], we will call the side most distant from the given angle the <strong>opposite side<\/strong> from angle [latex]t[\/latex]. And instead of [latex]1[\/latex], we will call the side of a right triangle opposite the right angle the <strong>hypotenuse<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>labeling right triangle sides<\/h3>\n<p><span id=\"fs-id1165137465030\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003732\/CNX_Precalc_Figure_05_04_0022.jpg\" alt=\"A right triangle with hypotenuse, opposite, and adjacent sides labeled.\" \/><\/span><\/p>\n<p style=\"text-align: center;\">The sides of a right triangle in relation to angle [latex]t[\/latex].<\/p>\n<\/section>\n<h2>Understanding Right Triangle Relationships<\/h2>\n<p>Given a right triangle with an acute angle of [latex]t[\/latex],<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&\\sin \\left(t\\right)=\\frac{\\text{opposite}}{\\text{hypotenuse}} \\\\ &\\cos \\left(t\\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}} \\\\ &\\tan \\left(t\\right)=\\frac{\\text{opposite}}{\\text{adjacent}} \\end{align}[\/latex]<\/div>\n<p>A common mnemonic for remembering these relationships is SOH-CAH-TOA, formed from the first letters of &#8220;<u>S<\/u>ine is <u>o<\/u>pposite over <u>h<\/u>ypotenuse, <u>C<\/u>osine is <u>a<\/u>djacent over <u>h<\/u>ypotenuse, <u>T<\/u>angent is <u>o<\/u>pposite over <u>a<\/u>djacent.&#8221;<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.<\/strong><\/p>\n<ol>\n<li>Find the sine as the ratio of the opposite side to the hypotenuse.<\/li>\n<li>Find the cosine as the ratio of the adjacent side to the hypotenuse.<\/li>\n<li>Find the tangent is the ratio of the opposite side to the adjacent side.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the triangle, find the value of [latex]\\cos \\alpha[\/latex].<span id=\"fs-id1165137414609\"><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\/27003734\/CNX_Precalc_Figure_05_04_0032.jpg\" alt=\"A right triangle with sid lengths of 8, 15, and 17. Angle alpha also labeled.\" width=\"487\" height=\"188\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880775\">Show Solution<\/button><\/p>\n<div id=\"q880775\" class=\"hidden-answer\" style=\"display: none\">The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\cos \\left(\\alpha \\right)=\\frac{\\text{adjacent}}{\\text{hypotenuse}} =\\frac{15}{17} \\end{align}[\/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>Given the triangle, find the value of [latex]\\text{sin}t[\/latex].<span id=\"fs-id1165135191134\"><br \/>\n<\/span><\/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\/27003736\/CNX_Precalc_Figure_05_04_0042.jpg\" alt=\"A right triangle with sides of 7, 24, and 25. Also labeled is angle t.\" width=\"487\" height=\"180\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q261765\">Show Solution<\/button><\/p>\n<div id=\"q261765\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{7}{25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":5,"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 Triangles\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Ujyl_zQw2zE\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Cofunction Identities\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/_gkuml--4_Q\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Example: Determine the Length of a Side of a Right Triangle Using a Trig Equation\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/8jU2R3BuR5E\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":221,"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 Triangles","author":"Mathispower4u","organization":"","url":"https:\/\/youtu.be\/Ujyl_zQw2zE","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Cofunction Identities","author":"Mathispower4u","organization":"","url":"https:\/\/youtu.be\/_gkuml--4_Q","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Example: Determine the Length of a Side of a Right Triangle Using a Trig Equation","author":"Mathispower4u","organization":"","url":"https:\/\/youtu.be\/8jU2R3BuR5E","project":"","license":"arr","license_terms":"Standard YouTube License"}],"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\/182"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions"}],"predecessor-version":[{"id":4729,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions\/4729"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/221"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/182\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=182"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=182"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=182"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}