{"id":1558,"date":"2025-07-25T03:41:15","date_gmt":"2025-07-25T03:41:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1558"},"modified":"2026-03-12T06:38:14","modified_gmt":"2026-03-12T06:38:14","slug":"right-triangle-trigonometry-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/right-triangle-trigonometry-fresh-take\/","title":{"raw":"Right Triangle Trigonometry: Fresh Take","rendered":"Right Triangle Trigonometry: Fresh Take"},"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>Evaluating Trigonometric Functions with Right Triangles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nRight triangles give us a straightforward way to define and evaluate trigonometric functions. By labeling the sides relative to an acute angle [latex]\\theta[\/latex]\u2014opposite, adjacent, and hypotenuse\u2014we can express sine, cosine, tangent, and their reciprocals as ratios of side lengths. This geometric perspective is often the first way students encounter trigonometry and is especially useful for evaluating trig functions of special angles like [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Right Triangles<\/strong>\r\n<ol>\r\n \t<li data-start=\"659\" data-end=\"1154\">\r\n<p data-start=\"662\" data-end=\"690\"><strong data-start=\"662\" data-end=\"688\">Recall the Definitions<\/strong><\/p>\r\n\r\n<ul data-start=\"694\" data-end=\"1154\">\r\n \t<li data-start=\"694\" data-end=\"768\">\r\n<p data-start=\"696\" data-end=\"768\">[latex]\\sin\\theta = \\dfrac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"772\" data-end=\"846\">\r\n<p data-start=\"774\" data-end=\"846\">[latex]\\cos\\theta = \\dfrac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"850\" data-end=\"922\">\r\n<p data-start=\"852\" data-end=\"922\">[latex]\\tan\\theta = \\dfrac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"926\" data-end=\"1000\">\r\n<p data-start=\"928\" data-end=\"1000\">[latex]\\csc\\theta = \\dfrac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1004\" data-end=\"1078\">\r\n<p data-start=\"1006\" data-end=\"1078\">[latex]\\sec\\theta = \\dfrac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1082\" data-end=\"1154\">\r\n<p data-start=\"1084\" data-end=\"1154\">[latex]\\cot\\theta = \\dfrac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1156\" data-end=\"1277\">\r\n<p data-start=\"1159\" data-end=\"1192\"><strong data-start=\"1159\" data-end=\"1190\">Use the Pythagorean Theorem<\/strong><\/p>\r\n\r\n<ul data-start=\"1196\" data-end=\"1277\">\r\n \t<li data-start=\"1196\" data-end=\"1277\">\r\n<p data-start=\"1198\" data-end=\"1277\">If two sides are known, find the third: [latex]a^{2} + b^{2} = c^{2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1279\" data-end=\"1649\">\r\n<p data-start=\"1654\" data-end=\"1688\"><strong data-start=\"1654\" data-end=\"1686\">Special Right Triangles Help<\/strong><\/p>\r\n\r\n<ul data-start=\"1692\" data-end=\"2058\">\r\n \t<li data-start=\"1692\" data-end=\"1795\">\r\n<p data-start=\"1694\" data-end=\"1795\">[latex]45^\\circ-45^\\circ-90^\\circ[\/latex] triangle: legs = 1, hypotenuse = [latex]\\sqrt{2}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1799\" data-end=\"1925\">\r\n<p data-start=\"1801\" data-end=\"1925\">[latex]30^\\circ-60^\\circ-90^\\circ[\/latex] triangle: shorter leg = 1, longer leg = [latex]\\sqrt{3}[\/latex], hypotenuse = 2.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1929\" data-end=\"2058\">\r\n<p data-start=\"1931\" data-end=\"2058\">Use these ratios to evaluate trig functions of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"2060\" data-end=\"2239\">\r\n<p data-start=\"2063\" data-end=\"2089\"><strong data-start=\"2063\" data-end=\"2087\">Remember the Context<\/strong><\/p>\r\n\r\n<ul data-start=\"2093\" data-end=\"2239\">\r\n \t<li data-start=\"2093\" data-end=\"2239\">\r\n<p data-start=\"2095\" data-end=\"2239\">Right-triangle definitions only apply to <strong data-start=\"2136\" data-end=\"2152\">acute angles<\/strong> inside right triangles, but they extend naturally to the unit circle for all angles.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A right triangle has an acute angle [latex]\\theta[\/latex] with opposite side 8 and adjacent side 15. Find all six trigonometric functions of [latex]\\theta[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"138091\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138091\"]Solution:<\/p>\r\nFirst, find the hypotenuse using the Pythagorean Theorem:\r\n[latex] \\begin{aligned} c^2 &amp;= 8^2 + 15^2 \\\\ c^2 &amp;= 64 + 225 \\\\ c^2 &amp;= 289 \\\\ c &amp;= 17 \\end{aligned} [\/latex]\r\n\r\nNow use the definitions with opposite = 8, adjacent = 15, and hypotenuse = 17:\r\n[latex] \\begin{aligned} \\sin\\theta &amp;= \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{8}{17} \\\\ \\cos\\theta &amp;= \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{15}{17} \\\\ \\tan\\theta &amp;= \\frac{\\text{opposite}}{\\text{adjacent}} = \\frac{8}{15} \\\\ \\csc\\theta &amp;= \\frac{\\text{hypotenuse}}{\\text{opposite}} = \\frac{17}{8} \\\\ \\sec\\theta &amp;= \\frac{\\text{hypotenuse}}{\\text{adjacent}} = \\frac{17}{15} \\\\ \\cot\\theta &amp;= \\frac{\\text{adjacent}}{\\text{opposite}} = \\frac{15}{8} \\end{aligned} [\/latex][\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcbhgah-Ujyl_zQw2zE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ujyl_zQw2zE?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hfcbhgah-Ujyl_zQw2zE\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661395&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hfcbhgah-Ujyl_zQw2zE&vembed=0&video_id=Ujyl_zQw2zE&video_target=tpm-plugin-hfcbhgah-Ujyl_zQw2zE'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Introduction+to+Trigonometric+Functions+Using+Triangles_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Trigonometric Functions Using Triangles\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Cofunctions of Complementary Angles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"64\" data-end=\"506\">Cofunction identities show that the trig function of an angle equals the cofunction of its complement. Since two acute angles in a right triangle add up to [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]), the sine of one angle equals the cosine of the other, tangent pairs with cotangent, and secant pairs with cosecant. These relationships are useful for simplifying expressions and recognizing equivalences between trig values.<\/p>\r\n<p data-start=\"508\" data-end=\"540\">The cofunction identities are:<\/p>\r\n\r\n<ul data-start=\"542\" data-end=\"1013\">\r\n \t<li data-start=\"542\" data-end=\"678\">\r\n<p data-start=\"544\" data-end=\"678\">[latex]\\sin\\theta = \\cos\\left(90^\\circ - \\theta\\right)[\/latex] or [latex]\\sin\\theta = \\cos\\left(\\dfrac{\\pi}{2}-\\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"679\" data-end=\"745\">\r\n<p data-start=\"681\" data-end=\"745\">[latex]\\cos\\theta = \\sin\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"746\" data-end=\"812\">\r\n<p data-start=\"748\" data-end=\"812\">[latex]\\tan\\theta = \\cot\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"813\" data-end=\"879\">\r\n<p data-start=\"815\" data-end=\"879\">[latex]\\cot\\theta = \\tan\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"880\" data-end=\"946\">\r\n<p data-start=\"882\" data-end=\"946\">[latex]\\sec\\theta = \\csc\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"947\" data-end=\"1013\">\r\n<p data-start=\"949\" data-end=\"1013\">[latex]\\csc\\theta = \\sec\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Cofunctions<\/strong>\r\n<ol data-start=\"1057\" data-end=\"1929\">\r\n \t<li data-start=\"1057\" data-end=\"1272\">\r\n<p data-start=\"1060\" data-end=\"1086\"><strong data-start=\"1060\" data-end=\"1084\">Think Right Triangle<\/strong><\/p>\r\n\r\n<ul data-start=\"1090\" data-end=\"1272\">\r\n \t<li data-start=\"1090\" data-end=\"1206\">\r\n<p data-start=\"1092\" data-end=\"1206\">In a right triangle, if one acute angle is [latex]\\theta[\/latex], the other is [latex]90^\\circ - \\theta[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1210\" data-end=\"1272\">\r\n<p data-start=\"1212\" data-end=\"1272\">The sine of one equals the cosine of the other, and so on.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1274\" data-end=\"1374\">\r\n<p data-start=\"1277\" data-end=\"1301\"><strong data-start=\"1277\" data-end=\"1299\">Memorize the Pairs<\/strong><\/p>\r\n\r\n<ul data-start=\"1776\" data-end=\"1929\">\r\n \t<li data-start=\"1305\" data-end=\"1322\">\r\n<p data-start=\"1307\" data-end=\"1322\">Sine \u2194 Cosine<\/p>\r\n<\/li>\r\n \t<li data-start=\"1326\" data-end=\"1349\">\r\n<p data-start=\"1328\" data-end=\"1349\">Tangent \u2194 Cotangent<\/p>\r\n<\/li>\r\n \t<li data-start=\"1353\" data-end=\"1374\">\r\n<p data-start=\"1355\" data-end=\"1374\">Secant \u2194 Cosecant<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Evaluate [latex]\\sin(25\u00b0)[\/latex] if you know that [latex]\\cos(65\u00b0) = 0.4226[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"912023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"912023\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solution:<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Notice that [latex]25\u00b0 + 65\u00b0 = 90\u00b0[\/latex], so these angles are complementary.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">By the cofunction identity: [latex] \\begin{aligned} \\sin(25\u00b0) &amp;= \\cos(90\u00b0 - 25\u00b0) \\\\ &amp;= \\cos(65\u00b0) \\\\ &amp;= 0.4226 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dhbffgcc-_gkuml--4_Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_gkuml--4_Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dhbffgcc-_gkuml--4_Q\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661396&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dhbffgcc-_gkuml--4_Q&vembed=0&video_id=_gkuml--4_Q&video_target=tpm-plugin-dhbffgcc-_gkuml--4_Q'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Cofunction+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCofunction Identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Trigonometric Functions of Any Angle<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"492\" data-end=\"849\">The definitions of trigonometric functions extend beyond right triangles to <strong data-start=\"568\" data-end=\"581\">any angle<\/strong> by using the unit circle or coordinates in the Cartesian plane. For an angle [latex]\\theta[\/latex] drawn in standard position with a point [latex](x,y)[\/latex] on its terminal side and distance [latex]r=\\sqrt{x^{2}+y^{2}}[\/latex], the trig functions are defined as:<\/p>\r\n\r\n<ul data-start=\"851\" data-end=\"1184\">\r\n \t<li data-start=\"851\" data-end=\"895\">\r\n<p data-start=\"853\" data-end=\"895\">[latex]\\sin\\theta = \\dfrac{y}{r}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"896\" data-end=\"940\">\r\n<p data-start=\"898\" data-end=\"940\">[latex]\\cos\\theta = \\dfrac{x}{r}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"941\" data-end=\"1001\">\r\n<p data-start=\"943\" data-end=\"1001\">[latex]\\tan\\theta = \\dfrac{y}{x}, \\quad x \\neq 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1002\" data-end=\"1062\">\r\n<p data-start=\"1004\" data-end=\"1062\">[latex]\\csc\\theta = \\dfrac{r}{y}, \\quad y \\neq 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1063\" data-end=\"1123\">\r\n<p data-start=\"1065\" data-end=\"1123\">[latex]\\sec\\theta = \\dfrac{r}{x}, \\quad x \\neq 0[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1124\" data-end=\"1184\">\r\n<p data-start=\"1126\" data-end=\"1184\">[latex]\\cot\\theta = \\dfrac{x}{y}, \\quad y \\neq 0[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"1186\" data-end=\"1308\">This framework works for all real angles, not just acute ones, and naturally incorporates negative values and quadrants.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Definitions for Any Angle<\/strong>\r\n<ol data-start=\"1366\" data-end=\"2422\">\r\n \t<li data-start=\"1366\" data-end=\"1508\">\r\n<p data-start=\"1369\" data-end=\"1397\"><strong data-start=\"1369\" data-end=\"1395\">Unit Circle Connection<\/strong><\/p>\r\n\r\n<ul data-start=\"1401\" data-end=\"1508\">\r\n \t<li data-start=\"1401\" data-end=\"1508\">\r\n<p data-start=\"1403\" data-end=\"1508\">On the unit circle, [latex]r=1[\/latex], so [latex]\\sin\\theta=y[\/latex] and [latex]\\cos\\theta=x[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1510\" data-end=\"1735\">\r\n<p data-start=\"1513\" data-end=\"1543\"><strong data-start=\"1513\" data-end=\"1541\">Signs Depend on Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"1547\" data-end=\"1735\">\r\n \t<li data-start=\"1547\" data-end=\"1586\">\r\n<p data-start=\"1549\" data-end=\"1586\">Quadrant I: all functions positive.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1590\" data-end=\"1634\">\r\n<p data-start=\"1592\" data-end=\"1634\">Quadrant II: sine and cosecant positive.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1638\" data-end=\"1687\">\r\n<p data-start=\"1640\" data-end=\"1687\">Quadrant III: tangent and cotangent positive.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1691\" data-end=\"1735\">\r\n<p data-start=\"1693\" data-end=\"1735\">Quadrant IV: cosine and secant positive.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"2237\" data-end=\"2422\">\r\n<p data-start=\"2240\" data-end=\"2269\"><strong data-start=\"2240\" data-end=\"2267\">Always Reduce to Ratios<\/strong><\/p>\r\n\r\n<ul data-start=\"2273\" data-end=\"2422\">\r\n \t<li data-start=\"2273\" data-end=\"2422\">\r\n<p data-start=\"2275\" data-end=\"2422\">No matter the quadrant, definitions reduce trig to [latex]\\dfrac{x}{r}[\/latex] and [latex]\\dfrac{y}{r}[\/latex], with signs handled automatically.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The terminal side of angle [latex]\\theta[\/latex] in standard position passes through the point [latex](-5, 12)[\/latex]. Find [latex]\\sin\\theta[\/latex], [latex]\\cos\\theta[\/latex], and [latex]\\tan\\theta[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"957296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"957296\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solution:<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">First, find [latex]r[\/latex] using the distance formula: [latex] \\begin{aligned} r &amp;= \\sqrt{x^2 + y^2} \\\\ &amp;= \\sqrt{(-5)^2 + 12^2} \\\\ &amp;= \\sqrt{25 + 144} \\\\ &amp;= \\sqrt{169} \\\\ &amp;= 13 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Now apply the definitions with [latex]x = -5[\/latex], [latex]y = 12[\/latex], and [latex]r = 13[\/latex]:<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\begin{aligned} \\sin\\theta &amp;= \\frac{y}{r} = \\frac{12}{13} \\\\ \\cos\\theta &amp;= \\frac{x}{r} = \\frac{-5}{13} = -\\frac{5}{13} \\\\ \\tan\\theta &amp;= \\frac{y}{x} = \\frac{12}{-5} = -\\frac{12}{5} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Notice that [latex]\\theta[\/latex] is in Quadrant II (since [latex]x &lt; 0[\/latex] and [latex]y &gt; 0[\/latex]), where sine is positive but cosine and tangent are negative.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-afddfcca-8jU2R3BuR5E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8jU2R3BuR5E?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-afddfcca-8jU2R3BuR5E\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661397&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-afddfcca-8jU2R3BuR5E&vembed=0&video_id=8jU2R3BuR5E&video_target=tpm-plugin-afddfcca-8jU2R3BuR5E'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Determine+the+Length+of+a+Side+of+a+Right+Triangle+Using+a+Trig+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine the Length of a Side of a Right Triangle Using a Trig Equation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Applied Problems with Right Triangle Trigonometry<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nRight triangle trigonometry helps us solve real-world problems that involve heights, distances, and angles. By modeling a situation as a right triangle, we can use sine, cosine, tangent, and their reciprocals to connect side lengths with angles. These tools are especially useful for applications such as measuring building heights, finding the distance across rivers, or calculating the angle of elevation or depression.\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Solving Applied Problems<\/strong>\r\n<ol>\r\n \t<li data-start=\"552\" data-end=\"695\">\r\n<p data-start=\"555\" data-end=\"575\"><strong data-start=\"555\" data-end=\"573\">Draw a Diagram<\/strong><\/p>\r\n\r\n<ul data-start=\"579\" data-end=\"695\">\r\n \t<li data-start=\"579\" data-end=\"623\">\r\n<p data-start=\"581\" data-end=\"623\">Sketch the scenario as a right triangle.<\/p>\r\n<\/li>\r\n \t<li data-start=\"627\" data-end=\"695\">\r\n<p data-start=\"629\" data-end=\"695\">Label the known sides and angles, and mark the unknown quantity.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"697\" data-end=\"898\">\r\n<p data-start=\"700\" data-end=\"736\"><strong data-start=\"700\" data-end=\"734\">Identify the Function You Need<\/strong><\/p>\r\n\r\n<ul data-start=\"740\" data-end=\"898\">\r\n \t<li data-start=\"740\" data-end=\"789\">\r\n<p data-start=\"742\" data-end=\"789\">Use sine if you know opposite and hypotenuse.<\/p>\r\n<\/li>\r\n \t<li data-start=\"793\" data-end=\"844\">\r\n<p data-start=\"795\" data-end=\"844\">Use cosine if you know adjacent and hypotenuse.<\/p>\r\n<\/li>\r\n \t<li data-start=\"848\" data-end=\"898\">\r\n<p data-start=\"850\" data-end=\"898\">Use tangent if you know opposite and adjacent.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"900\" data-end=\"1112\">\r\n<p data-start=\"903\" data-end=\"938\"><strong data-start=\"903\" data-end=\"936\">Angle of Elevation\/Depression<\/strong><\/p>\r\n\r\n<ul data-start=\"942\" data-end=\"1112\">\r\n \t<li data-start=\"942\" data-end=\"995\">\r\n<p data-start=\"944\" data-end=\"995\">Elevation: angle measured upward from horizontal.<\/p>\r\n<\/li>\r\n \t<li data-start=\"999\" data-end=\"1055\">\r\n<p data-start=\"1001\" data-end=\"1055\">Depression: angle measured downward from horizontal.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1059\" data-end=\"1112\">\r\n<p data-start=\"1061\" data-end=\"1112\">Both are drawn with respect to a horizontal line.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1114\" data-end=\"1334\">\r\n<p data-start=\"1339\" data-end=\"1373\"><strong data-start=\"1339\" data-end=\"1371\">Use Inverse Trig When Needed<\/strong><\/p>\r\n\r\n<ul data-start=\"1377\" data-end=\"1517\">\r\n \t<li data-start=\"1377\" data-end=\"1517\">\r\n<p data-start=\"1379\" data-end=\"1517\">If side lengths are known but the angle is unknown, use [latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], or [latex]\\tan^{-1}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1519\" data-end=\"1727\">\r\n<p data-start=\"1522\" data-end=\"1551\"><strong data-start=\"1522\" data-end=\"1549\">Check Units and Context<\/strong><\/p>\r\n\r\n<ul data-start=\"1555\" data-end=\"1727\">\r\n \t<li data-start=\"1555\" data-end=\"1644\">\r\n<p data-start=\"1557\" data-end=\"1644\">Make sure answers are expressed with appropriate units (feet, meters, degrees, etc.).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1648\" data-end=\"1727\">\r\n<p data-start=\"1650\" data-end=\"1727\">Interpret the solution in the context of the problem, not just as a number.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A surveyor stands 50 feet from the base of a building. The angle of elevation to the top of the building is [latex]68\u00b0[\/latex]. How tall is the building?<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"12990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"12990\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solution:<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Draw a diagram showing a right triangle where:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The horizontal distance (adjacent side) is 50 feet<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The height of the building (opposite side) is unknown<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The angle of elevation is [latex]68\u00b0[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Use tangent since we have adjacent and need opposite: [latex] \\begin{aligned} \\tan(68\u00b0) &amp;= \\frac{\\text{height}}{50} \\\\ \\text{height} &amp;= 50 \\cdot \\tan(68\u00b0) \\\\ \\text{height} &amp;= 50 \\cdot 2.4751 \\\\ \\text{height} &amp;\\approx 123.8 \\text{ feet} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The building is approximately 124 feet tall.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>","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>Evaluating Trigonometric Functions with Right Triangles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Right triangles give us a straightforward way to define and evaluate trigonometric functions. By labeling the sides relative to an acute angle [latex]\\theta[\/latex]\u2014opposite, adjacent, and hypotenuse\u2014we can express sine, cosine, tangent, and their reciprocals as ratios of side lengths. This geometric perspective is often the first way students encounter trigonometry and is especially useful for evaluating trig functions of special angles like [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Right Triangles<\/strong><\/p>\n<ol>\n<li data-start=\"659\" data-end=\"1154\">\n<p data-start=\"662\" data-end=\"690\"><strong data-start=\"662\" data-end=\"688\">Recall the Definitions<\/strong><\/p>\n<ul data-start=\"694\" data-end=\"1154\">\n<li data-start=\"694\" data-end=\"768\">\n<p data-start=\"696\" data-end=\"768\">[latex]\\sin\\theta = \\dfrac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"772\" data-end=\"846\">\n<p data-start=\"774\" data-end=\"846\">[latex]\\cos\\theta = \\dfrac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"850\" data-end=\"922\">\n<p data-start=\"852\" data-end=\"922\">[latex]\\tan\\theta = \\dfrac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"926\" data-end=\"1000\">\n<p data-start=\"928\" data-end=\"1000\">[latex]\\csc\\theta = \\dfrac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1004\" data-end=\"1078\">\n<p data-start=\"1006\" data-end=\"1078\">[latex]\\sec\\theta = \\dfrac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1082\" data-end=\"1154\">\n<p data-start=\"1084\" data-end=\"1154\">[latex]\\cot\\theta = \\dfrac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1156\" data-end=\"1277\">\n<p data-start=\"1159\" data-end=\"1192\"><strong data-start=\"1159\" data-end=\"1190\">Use the Pythagorean Theorem<\/strong><\/p>\n<ul data-start=\"1196\" data-end=\"1277\">\n<li data-start=\"1196\" data-end=\"1277\">\n<p data-start=\"1198\" data-end=\"1277\">If two sides are known, find the third: [latex]a^{2} + b^{2} = c^{2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1279\" data-end=\"1649\">\n<p data-start=\"1654\" data-end=\"1688\"><strong data-start=\"1654\" data-end=\"1686\">Special Right Triangles Help<\/strong><\/p>\n<ul data-start=\"1692\" data-end=\"2058\">\n<li data-start=\"1692\" data-end=\"1795\">\n<p data-start=\"1694\" data-end=\"1795\">[latex]45^\\circ-45^\\circ-90^\\circ[\/latex] triangle: legs = 1, hypotenuse = [latex]\\sqrt{2}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1799\" data-end=\"1925\">\n<p data-start=\"1801\" data-end=\"1925\">[latex]30^\\circ-60^\\circ-90^\\circ[\/latex] triangle: shorter leg = 1, longer leg = [latex]\\sqrt{3}[\/latex], hypotenuse = 2.<\/p>\n<\/li>\n<li data-start=\"1929\" data-end=\"2058\">\n<p data-start=\"1931\" data-end=\"2058\">Use these ratios to evaluate trig functions of [latex]30^\\circ[\/latex], [latex]45^\\circ[\/latex], and [latex]60^\\circ[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2060\" data-end=\"2239\">\n<p data-start=\"2063\" data-end=\"2089\"><strong data-start=\"2063\" data-end=\"2087\">Remember the Context<\/strong><\/p>\n<ul data-start=\"2093\" data-end=\"2239\">\n<li data-start=\"2093\" data-end=\"2239\">\n<p data-start=\"2095\" data-end=\"2239\">Right-triangle definitions only apply to <strong data-start=\"2136\" data-end=\"2152\">acute angles<\/strong> inside right triangles, but they extend naturally to the unit circle for all angles.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A right triangle has an acute angle [latex]\\theta[\/latex] with opposite side 8 and adjacent side 15. Find all six trigonometric functions of [latex]\\theta[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q138091\">Show Solution<\/button><\/p>\n<div id=\"q138091\" class=\"hidden-answer\" style=\"display: none\">Solution:<\/p>\n<p>First, find the hypotenuse using the Pythagorean Theorem:<br \/>\n[latex]\\begin{aligned} c^2 &= 8^2 + 15^2 \\\\ c^2 &= 64 + 225 \\\\ c^2 &= 289 \\\\ c &= 17 \\end{aligned}[\/latex]<\/p>\n<p>Now use the definitions with opposite = 8, adjacent = 15, and hypotenuse = 17:<br \/>\n[latex]\\begin{aligned} \\sin\\theta &= \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{8}{17} \\\\ \\cos\\theta &= \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{15}{17} \\\\ \\tan\\theta &= \\frac{\\text{opposite}}{\\text{adjacent}} = \\frac{8}{15} \\\\ \\csc\\theta &= \\frac{\\text{hypotenuse}}{\\text{opposite}} = \\frac{17}{8} \\\\ \\sec\\theta &= \\frac{\\text{hypotenuse}}{\\text{adjacent}} = \\frac{17}{15} \\\\ \\cot\\theta &= \\frac{\\text{adjacent}}{\\text{opposite}} = \\frac{15}{8} \\end{aligned}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcbhgah-Ujyl_zQw2zE\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ujyl_zQw2zE?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hfcbhgah-Ujyl_zQw2zE\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661395&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hfcbhgah-Ujyl_zQw2zE&#38;vembed=0&#38;video_id=Ujyl_zQw2zE&#38;video_target=tpm-plugin-hfcbhgah-Ujyl_zQw2zE\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Introduction+to+Trigonometric+Functions+Using+Triangles_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Trigonometric Functions Using Triangles\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Cofunctions of Complementary Angles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"64\" data-end=\"506\">Cofunction identities show that the trig function of an angle equals the cofunction of its complement. Since two acute angles in a right triangle add up to [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]), the sine of one angle equals the cosine of the other, tangent pairs with cotangent, and secant pairs with cosecant. These relationships are useful for simplifying expressions and recognizing equivalences between trig values.<\/p>\n<p data-start=\"508\" data-end=\"540\">The cofunction identities are:<\/p>\n<ul data-start=\"542\" data-end=\"1013\">\n<li data-start=\"542\" data-end=\"678\">\n<p data-start=\"544\" data-end=\"678\">[latex]\\sin\\theta = \\cos\\left(90^\\circ - \\theta\\right)[\/latex] or [latex]\\sin\\theta = \\cos\\left(\\dfrac{\\pi}{2}-\\theta\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"679\" data-end=\"745\">\n<p data-start=\"681\" data-end=\"745\">[latex]\\cos\\theta = \\sin\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"746\" data-end=\"812\">\n<p data-start=\"748\" data-end=\"812\">[latex]\\tan\\theta = \\cot\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"813\" data-end=\"879\">\n<p data-start=\"815\" data-end=\"879\">[latex]\\cot\\theta = \\tan\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"880\" data-end=\"946\">\n<p data-start=\"882\" data-end=\"946\">[latex]\\sec\\theta = \\csc\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\n<\/li>\n<li data-start=\"947\" data-end=\"1013\">\n<p data-start=\"949\" data-end=\"1013\">[latex]\\csc\\theta = \\sec\\left(90^\\circ - \\theta\\right)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Cofunctions<\/strong><\/p>\n<ol data-start=\"1057\" data-end=\"1929\">\n<li data-start=\"1057\" data-end=\"1272\">\n<p data-start=\"1060\" data-end=\"1086\"><strong data-start=\"1060\" data-end=\"1084\">Think Right Triangle<\/strong><\/p>\n<ul data-start=\"1090\" data-end=\"1272\">\n<li data-start=\"1090\" data-end=\"1206\">\n<p data-start=\"1092\" data-end=\"1206\">In a right triangle, if one acute angle is [latex]\\theta[\/latex], the other is [latex]90^\\circ - \\theta[\/latex].<\/p>\n<\/li>\n<li data-start=\"1210\" data-end=\"1272\">\n<p data-start=\"1212\" data-end=\"1272\">The sine of one equals the cosine of the other, and so on.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1274\" data-end=\"1374\">\n<p data-start=\"1277\" data-end=\"1301\"><strong data-start=\"1277\" data-end=\"1299\">Memorize the Pairs<\/strong><\/p>\n<ul data-start=\"1776\" data-end=\"1929\">\n<li data-start=\"1305\" data-end=\"1322\">\n<p data-start=\"1307\" data-end=\"1322\">Sine \u2194 Cosine<\/p>\n<\/li>\n<li data-start=\"1326\" data-end=\"1349\">\n<p data-start=\"1328\" data-end=\"1349\">Tangent \u2194 Cotangent<\/p>\n<\/li>\n<li data-start=\"1353\" data-end=\"1374\">\n<p data-start=\"1355\" data-end=\"1374\">Secant \u2194 Cosecant<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Evaluate [latex]\\sin(25\u00b0)[\/latex] if you know that [latex]\\cos(65\u00b0) = 0.4226[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q912023\">Show Solution<\/button><\/p>\n<div id=\"q912023\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><strong>Solution:<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Notice that [latex]25\u00b0 + 65\u00b0 = 90\u00b0[\/latex], so these angles are complementary.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">By the cofunction identity: [latex]\\begin{aligned} \\sin(25\u00b0) &= \\cos(90\u00b0 - 25\u00b0) \\\\ &= \\cos(65\u00b0) \\\\ &= 0.4226 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dhbffgcc-_gkuml--4_Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_gkuml--4_Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dhbffgcc-_gkuml--4_Q\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661396&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dhbffgcc-_gkuml--4_Q&#38;vembed=0&#38;video_id=_gkuml--4_Q&#38;video_target=tpm-plugin-dhbffgcc-_gkuml--4_Q\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Cofunction+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCofunction Identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Trigonometric Functions of Any Angle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"492\" data-end=\"849\">The definitions of trigonometric functions extend beyond right triangles to <strong data-start=\"568\" data-end=\"581\">any angle<\/strong> by using the unit circle or coordinates in the Cartesian plane. For an angle [latex]\\theta[\/latex] drawn in standard position with a point [latex](x,y)[\/latex] on its terminal side and distance [latex]r=\\sqrt{x^{2}+y^{2}}[\/latex], the trig functions are defined as:<\/p>\n<ul data-start=\"851\" data-end=\"1184\">\n<li data-start=\"851\" data-end=\"895\">\n<p data-start=\"853\" data-end=\"895\">[latex]\\sin\\theta = \\dfrac{y}{r}[\/latex]<\/p>\n<\/li>\n<li data-start=\"896\" data-end=\"940\">\n<p data-start=\"898\" data-end=\"940\">[latex]\\cos\\theta = \\dfrac{x}{r}[\/latex]<\/p>\n<\/li>\n<li data-start=\"941\" data-end=\"1001\">\n<p data-start=\"943\" data-end=\"1001\">[latex]\\tan\\theta = \\dfrac{y}{x}, \\quad x \\neq 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"1002\" data-end=\"1062\">\n<p data-start=\"1004\" data-end=\"1062\">[latex]\\csc\\theta = \\dfrac{r}{y}, \\quad y \\neq 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"1063\" data-end=\"1123\">\n<p data-start=\"1065\" data-end=\"1123\">[latex]\\sec\\theta = \\dfrac{r}{x}, \\quad x \\neq 0[\/latex]<\/p>\n<\/li>\n<li data-start=\"1124\" data-end=\"1184\">\n<p data-start=\"1126\" data-end=\"1184\">[latex]\\cot\\theta = \\dfrac{x}{y}, \\quad y \\neq 0[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1186\" data-end=\"1308\">This framework works for all real angles, not just acute ones, and naturally incorporates negative values and quadrants.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Definitions for Any Angle<\/strong><\/p>\n<ol data-start=\"1366\" data-end=\"2422\">\n<li data-start=\"1366\" data-end=\"1508\">\n<p data-start=\"1369\" data-end=\"1397\"><strong data-start=\"1369\" data-end=\"1395\">Unit Circle Connection<\/strong><\/p>\n<ul data-start=\"1401\" data-end=\"1508\">\n<li data-start=\"1401\" data-end=\"1508\">\n<p data-start=\"1403\" data-end=\"1508\">On the unit circle, [latex]r=1[\/latex], so [latex]\\sin\\theta=y[\/latex] and [latex]\\cos\\theta=x[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1510\" data-end=\"1735\">\n<p data-start=\"1513\" data-end=\"1543\"><strong data-start=\"1513\" data-end=\"1541\">Signs Depend on Quadrant<\/strong><\/p>\n<ul data-start=\"1547\" data-end=\"1735\">\n<li data-start=\"1547\" data-end=\"1586\">\n<p data-start=\"1549\" data-end=\"1586\">Quadrant I: all functions positive.<\/p>\n<\/li>\n<li data-start=\"1590\" data-end=\"1634\">\n<p data-start=\"1592\" data-end=\"1634\">Quadrant II: sine and cosecant positive.<\/p>\n<\/li>\n<li data-start=\"1638\" data-end=\"1687\">\n<p data-start=\"1640\" data-end=\"1687\">Quadrant III: tangent and cotangent positive.<\/p>\n<\/li>\n<li data-start=\"1691\" data-end=\"1735\">\n<p data-start=\"1693\" data-end=\"1735\">Quadrant IV: cosine and secant positive.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2237\" data-end=\"2422\">\n<p data-start=\"2240\" data-end=\"2269\"><strong data-start=\"2240\" data-end=\"2267\">Always Reduce to Ratios<\/strong><\/p>\n<ul data-start=\"2273\" data-end=\"2422\">\n<li data-start=\"2273\" data-end=\"2422\">\n<p data-start=\"2275\" data-end=\"2422\">No matter the quadrant, definitions reduce trig to [latex]\\dfrac{x}{r}[\/latex] and [latex]\\dfrac{y}{r}[\/latex], with signs handled automatically.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The terminal side of angle [latex]\\theta[\/latex] in standard position passes through the point [latex](-5, 12)[\/latex]. Find [latex]\\sin\\theta[\/latex], [latex]\\cos\\theta[\/latex], and [latex]\\tan\\theta[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q957296\">Show Solution<\/button><\/p>\n<div id=\"q957296\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><strong>Solution:<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">First, find [latex]r[\/latex] using the distance formula: [latex]\\begin{aligned} r &= \\sqrt{x^2 + y^2} \\\\ &= \\sqrt{(-5)^2 + 12^2} \\\\ &= \\sqrt{25 + 144} \\\\ &= \\sqrt{169} \\\\ &= 13 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Now apply the definitions with [latex]x = -5[\/latex], [latex]y = 12[\/latex], and [latex]r = 13[\/latex]:<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{aligned} \\sin\\theta &= \\frac{y}{r} = \\frac{12}{13} \\\\ \\cos\\theta &= \\frac{x}{r} = \\frac{-5}{13} = -\\frac{5}{13} \\\\ \\tan\\theta &= \\frac{y}{x} = \\frac{12}{-5} = -\\frac{12}{5} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Notice that [latex]\\theta[\/latex] is in Quadrant II (since [latex]x < 0[\/latex] and [latex]y > 0[\/latex]), where sine is positive but cosine and tangent are negative.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-afddfcca-8jU2R3BuR5E\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8jU2R3BuR5E?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-afddfcca-8jU2R3BuR5E\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661397&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-afddfcca-8jU2R3BuR5E&#38;vembed=0&#38;video_id=8jU2R3BuR5E&#38;video_target=tpm-plugin-afddfcca-8jU2R3BuR5E\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Determine+the+Length+of+a+Side+of+a+Right+Triangle+Using+a+Trig+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Determine the Length of a Side of a Right Triangle Using a Trig Equation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Applied Problems with Right Triangle Trigonometry<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Right triangle trigonometry helps us solve real-world problems that involve heights, distances, and angles. By modeling a situation as a right triangle, we can use sine, cosine, tangent, and their reciprocals to connect side lengths with angles. These tools are especially useful for applications such as measuring building heights, finding the distance across rivers, or calculating the angle of elevation or depression.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Solving Applied Problems<\/strong><\/p>\n<ol>\n<li data-start=\"552\" data-end=\"695\">\n<p data-start=\"555\" data-end=\"575\"><strong data-start=\"555\" data-end=\"573\">Draw a Diagram<\/strong><\/p>\n<ul data-start=\"579\" data-end=\"695\">\n<li data-start=\"579\" data-end=\"623\">\n<p data-start=\"581\" data-end=\"623\">Sketch the scenario as a right triangle.<\/p>\n<\/li>\n<li data-start=\"627\" data-end=\"695\">\n<p data-start=\"629\" data-end=\"695\">Label the known sides and angles, and mark the unknown quantity.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"697\" data-end=\"898\">\n<p data-start=\"700\" data-end=\"736\"><strong data-start=\"700\" data-end=\"734\">Identify the Function You Need<\/strong><\/p>\n<ul data-start=\"740\" data-end=\"898\">\n<li data-start=\"740\" data-end=\"789\">\n<p data-start=\"742\" data-end=\"789\">Use sine if you know opposite and hypotenuse.<\/p>\n<\/li>\n<li data-start=\"793\" data-end=\"844\">\n<p data-start=\"795\" data-end=\"844\">Use cosine if you know adjacent and hypotenuse.<\/p>\n<\/li>\n<li data-start=\"848\" data-end=\"898\">\n<p data-start=\"850\" data-end=\"898\">Use tangent if you know opposite and adjacent.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"900\" data-end=\"1112\">\n<p data-start=\"903\" data-end=\"938\"><strong data-start=\"903\" data-end=\"936\">Angle of Elevation\/Depression<\/strong><\/p>\n<ul data-start=\"942\" data-end=\"1112\">\n<li data-start=\"942\" data-end=\"995\">\n<p data-start=\"944\" data-end=\"995\">Elevation: angle measured upward from horizontal.<\/p>\n<\/li>\n<li data-start=\"999\" data-end=\"1055\">\n<p data-start=\"1001\" data-end=\"1055\">Depression: angle measured downward from horizontal.<\/p>\n<\/li>\n<li data-start=\"1059\" data-end=\"1112\">\n<p data-start=\"1061\" data-end=\"1112\">Both are drawn with respect to a horizontal line.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1114\" data-end=\"1334\">\n<p data-start=\"1339\" data-end=\"1373\"><strong data-start=\"1339\" data-end=\"1371\">Use Inverse Trig When Needed<\/strong><\/p>\n<ul data-start=\"1377\" data-end=\"1517\">\n<li data-start=\"1377\" data-end=\"1517\">\n<p data-start=\"1379\" data-end=\"1517\">If side lengths are known but the angle is unknown, use [latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], or [latex]\\tan^{-1}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1519\" data-end=\"1727\">\n<p data-start=\"1522\" data-end=\"1551\"><strong data-start=\"1522\" data-end=\"1549\">Check Units and Context<\/strong><\/p>\n<ul data-start=\"1555\" data-end=\"1727\">\n<li data-start=\"1555\" data-end=\"1644\">\n<p data-start=\"1557\" data-end=\"1644\">Make sure answers are expressed with appropriate units (feet, meters, degrees, etc.).<\/p>\n<\/li>\n<li data-start=\"1648\" data-end=\"1727\">\n<p data-start=\"1650\" data-end=\"1727\">Interpret the solution in the context of the problem, not just as a number.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A surveyor stands 50 feet from the base of a building. The angle of elevation to the top of the building is [latex]68\u00b0[\/latex]. How tall is the building?<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q12990\">Show Solution<\/button><\/p>\n<div id=\"q12990\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><strong>Solution:<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Draw a diagram showing a right triangle where:<\/p>\n<ul class=\"&#091;li_&amp;&#093;:mb-0 &#091;li_&amp;&#093;:mt-1 &#091;li_&amp;&#093;:gap-1 &#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The horizontal distance (adjacent side) is 50 feet<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The height of the building (opposite side) is unknown<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The angle of elevation is [latex]68\u00b0[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Use tangent since we have adjacent and need opposite: [latex]\\begin{aligned} \\tan(68\u00b0) &= \\frac{\\text{height}}{50} \\\\ \\text{height} &= 50 \\cdot \\tan(68\u00b0) \\\\ \\text{height} &= 50 \\cdot 2.4751 \\\\ \\text{height} &\\approx 123.8 \\text{ feet} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The building is approximately 124 feet tall.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal 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