{"id":1525,"date":"2025-07-25T02:33:08","date_gmt":"2025-07-25T02:33:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1525"},"modified":"2026-03-12T06:42:51","modified_gmt":"2026-03-12T06:42:51","slug":"inverse-trigonometric-functions-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-trigonometric-functions-2\/","title":{"raw":"Inverse Trigonometric Functions: Fresh Take","rendered":"Inverse Trigonometric Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Understand the domain restrictions on inverse sine, cosine, and tangent<\/li>\r\n \t<li>Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.<\/li>\r\n \t<li>Use a calculator to evaluate inverse trigonometric functions.<\/li>\r\n \t<li>Use inverse trigonometric functions to solve right triangles.<\/li>\r\n \t<li>Find exact values of composite functions with inverse trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Domain Restrictions on Inverse Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nInverse trigonometric functions \u201cundo\u201d sine, cosine, and tangent. But since sine, cosine, and tangent are <strong data-start=\"311\" data-end=\"323\">periodic<\/strong>, they are not one-to-one over their entire domains. To make their inverses valid functions, we must <strong data-start=\"424\" data-end=\"447\">restrict the domain<\/strong> of each trig function to an interval where it is one-to-one and covers all possible outputs. These restricted intervals are chosen so the inverse functions give a <strong data-start=\"611\" data-end=\"640\">single, consistent answer<\/strong>.\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: Domain Restrictions for Inverse Trig<\/strong>\r\n<ol>\r\n \t<li data-start=\"706\" data-end=\"944\">\r\n<p data-start=\"709\" data-end=\"761\"><strong data-start=\"709\" data-end=\"759\">Inverse Sine ([latex]y = \\sin^{-1}(x)[\/latex])<\/strong><\/p>\r\n\r\n<ul data-start=\"765\" data-end=\"944\">\r\n \t<li data-start=\"765\" data-end=\"808\">\r\n<p data-start=\"767\" data-end=\"808\">Domain: [latex]-1 \\leq x \\leq 1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"812\" data-end=\"880\">\r\n<p data-start=\"814\" data-end=\"880\">Range: [latex]-\\dfrac{\\pi}{2} \\leq y \\leq \\dfrac{\\pi}{2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"884\" data-end=\"944\">\r\n<p data-start=\"886\" data-end=\"944\">Chosen because sine is one-to-one in Quadrants I and IV.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"949\" data-end=\"1003\"><strong data-start=\"949\" data-end=\"1001\">Inverse Cosine ([latex]y = \\cos^{-1}(x)[\/latex])<\/strong>\r\n<ul data-start=\"1007\" data-end=\"1163\">\r\n \t<li data-start=\"1007\" data-end=\"1050\">\r\n<p data-start=\"1009\" data-end=\"1050\">Domain: [latex]-1 \\leq x \\leq 1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1054\" data-end=\"1097\">\r\n<p data-start=\"1056\" data-end=\"1097\">Range: [latex]0 \\leq y \\leq \\pi[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1101\" data-end=\"1163\">\r\n<p data-start=\"1103\" data-end=\"1163\">Chosen because cosine is one-to-one in Quadrants I and II.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1168\" data-end=\"1223\"><strong data-start=\"1168\" data-end=\"1221\">Inverse Tangent ([latex]y = \\tan^{-1}(x)[\/latex])<\/strong>\r\n<ul data-start=\"1227\" data-end=\"1421\">\r\n \t<li data-start=\"1227\" data-end=\"1288\">\r\n<p data-start=\"1229\" data-end=\"1288\">Domain: all real numbers [latex](-\\infty, \\infty)[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1292\" data-end=\"1354\">\r\n<p data-start=\"1294\" data-end=\"1354\">Range: [latex]-\\dfrac{\\pi}{2} &lt; y &lt; \\dfrac{\\pi}{2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1358\" data-end=\"1421\">\r\n<p data-start=\"1360\" data-end=\"1421\">Chosen because tangent is one-to-one in Quadrants I and IV.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcfhhhc-YXWKpgmLgHk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YXWKpgmLgHk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hfcfhhhc-YXWKpgmLgHk\" 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=14661398&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hfcfhhhc-YXWKpgmLgHk&vembed=0&video_id=YXWKpgmLgHk&video_target=tpm-plugin-hfcfhhhc-YXWKpgmLgHk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Inverse+Trigonometric+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInverse Trigonometric Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Determine whether each value is in the domain of the given inverse function. If it is, find the exact value.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]\\sin^{-1}\\left(\\frac{3}{2}\\right)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\cos^{-1}\\left(-\\frac{1}{2}\\right)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]\\tan^{-1}(5)[\/latex]<\/p>\r\n[reveal-answer q=\"420918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"420918\"]\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) The domain of [latex]\\sin^{-1}(x)[\/latex] is [latex][-1, 1][\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]\\frac{3}{2} = 1.5 &gt; 1[\/latex], this value is not in the domain.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]\\sin^{-1}\\left(\\frac{3}{2}\\right)[\/latex] is undefined.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) The domain of [latex]\\cos^{-1}(x)[\/latex] is [latex][-1, 1][\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]-1 \\leq -\\frac{1}{2} \\leq 1[\/latex], this value is in the domain.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">We need an angle in [latex][0, \\pi][\/latex] whose cosine is [latex]-\\frac{1}{2}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\cos^{-1}\\left(-\\frac{1}{2}\\right) = \\frac{2\\pi}{3} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">(This is in Quadrant II, where cosine is negative)<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) The domain of [latex]\\tan^{-1}(x)[\/latex] is all real numbers [latex](-\\infty, \\infty)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since 5 is a real number, this value is in the domain.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]\\tan^{-1}(5) \\approx 1.3734[\/latex] radians (or about [latex]78.69\u00b0[\/latex])<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<h2>Finding Exact Values of Inverse Trigonometric Expressions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"366\" data-end=\"1071\">Inverse trig functions return a <strong data-start=\"484\" data-end=\"503\">principal angle<\/strong> whose trig value matches the input. Because sine, cosine, and tangent are periodic, their inverses must be restricted to certain ranges so they give a single, consistent answer. To find exact values, use the <strong data-start=\"712\" data-end=\"733\">special triangles<\/strong> ([latex]30^\\circ[\/latex]\u2013[latex]60^\\circ[\/latex]\u2013[latex]90^\\circ[\/latex] and [latex]45^\\circ[\/latex]\u2013[latex]45^\\circ[\/latex]\u2013[latex]90^\\circ[\/latex]) or the unit circle. For composite expressions like [latex]\\sin(\\cos^{-1} x)[\/latex] or [latex]\\tan(\\sin^{-1} x)[\/latex], build a right triangle from the inner inverse, then use it to evaluate the requested function \u2014 always checking the <strong data-start=\"1121\" data-end=\"1129\">sign<\/strong> based on the inverse\u2019s range.<\/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: Exact Value with Inverse Sine, Cosine, and Tangent<\/strong>\r\n<ol>\r\n \t<li data-start=\"1239\" data-end=\"1501\">\r\n<p data-start=\"1242\" data-end=\"1274\"><strong data-start=\"1242\" data-end=\"1272\">Know the Principal Ranges:<\/strong><\/p>\r\n\r\n<ul data-start=\"1278\" data-end=\"1501\">\r\n \t<li data-start=\"1278\" data-end=\"1358\">\r\n<p data-start=\"1280\" data-end=\"1358\">[latex]\\sin^{-1}(x) \\in \\left[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right][\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1362\" data-end=\"1417\">\r\n<p data-start=\"1364\" data-end=\"1417\">[latex]\\cos^{-1}(x) \\in \\left[0, \\pi\\right][\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1421\" data-end=\"1501\">\r\n<p data-start=\"1423\" data-end=\"1501\">[latex]\\tan^{-1}(x) \\in \\left(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right)[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1503\" data-end=\"1725\">\r\n<p data-start=\"1506\" data-end=\"1527\"><strong data-start=\"1506\" data-end=\"1525\">Special Angles:<\/strong><\/p>\r\n\r\n<ul data-start=\"1531\" data-end=\"1725\">\r\n \t<li data-start=\"1531\" data-end=\"1601\">\r\n<p data-start=\"1533\" data-end=\"1601\">[latex]\\sin^{-1}\\left(\\dfrac{1}{2}\\right)=\\dfrac{\\pi}{6}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1605\" data-end=\"1675\">\r\n<p data-start=\"1607\" data-end=\"1675\">[latex]\\cos^{-1}\\left(\\dfrac{1}{2}\\right)=\\dfrac{\\pi}{3}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1679\" data-end=\"1725\">\r\n<p data-start=\"1681\" data-end=\"1725\">[latex]\\tan^{-1}(1)=\\dfrac{\\pi}{4}[\/latex]<\/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]\">Find the exact value of each expression.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]\\sin^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\cos^{-1}(0)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]\\tan^{-1}(-1)[\/latex]<\/p>\r\n[reveal-answer q=\"829825\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"829825\"]\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) We need an angle [latex]\\theta[\/latex] in [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex] where [latex]\\sin\\theta = \\frac{\\sqrt{3}}{2}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From special angles: [latex]\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]\\frac{\\pi}{3}[\/latex] is in the range [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex]:<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\sin^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right) = \\frac{\\pi}{3} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) We need an angle [latex]\\theta[\/latex] in [latex][0, \\pi][\/latex] where [latex]\\cos\\theta = 0[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From the unit circle: [latex]\\cos\\left(\\frac{\\pi}{2}\\right) = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\cos^{-1}(0) = \\frac{\\pi}{2} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) We need an angle [latex]\\theta[\/latex] in [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex] where [latex]\\tan\\theta = -1[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From special angles: [latex]\\tan\\left(-\\frac{\\pi}{4}\\right) = -1[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]-\\frac{\\pi}{4}[\/latex] is in the range [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex]:<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex] \\tan^{-1}(-1) = -\\frac{\\pi}{4} [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<h2>Evaluating Inverse Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"366\" data-end=\"1071\">Inverse trigonometric functions allow us to find the angle when we know the value of sine, cosine, or tangent. Since most values do not correspond to \u201cspecial angles,\u201d we rely on a calculator for decimal approximations. The calculator gives the <strong data-start=\"678\" data-end=\"697\">principal value<\/strong> of the inverse function, which comes from the restricted range of each inverse. To get correct results, the most important step is setting the calculator to the right mode: degrees if the answer should be in degrees, or radians if the answer should be in radians.<\/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 a Calculator for Inverse Trig<\/strong>\r\n<ol>\r\n \t<li data-start=\"1025\" data-end=\"1158\">\r\n<p data-start=\"1028\" data-end=\"1054\"><strong data-start=\"1028\" data-end=\"1052\">Check the Mode First<\/strong><\/p>\r\n\r\n<ul data-start=\"1058\" data-end=\"1158\">\r\n \t<li data-start=\"1058\" data-end=\"1106\">\r\n<p data-start=\"1060\" data-end=\"1106\">Degree mode if you want an angle in degrees.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1110\" data-end=\"1158\">\r\n<p data-start=\"1112\" data-end=\"1158\">Radian mode if you want an angle in radians.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1160\" data-end=\"1395\">\r\n<p data-start=\"1163\" data-end=\"1189\"><strong data-start=\"1163\" data-end=\"1187\">Use the Inverse Keys<\/strong><\/p>\r\n\r\n<ul data-start=\"1193\" data-end=\"1395\">\r\n \t<li data-start=\"1193\" data-end=\"1305\">\r\n<p data-start=\"1195\" data-end=\"1305\">Most calculators label them as [latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], [latex]\\tan^{-1}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1309\" data-end=\"1395\">\r\n<p data-start=\"1311\" data-end=\"1395\">Enter the number first or press the function button first (depends on calculator).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1397\" data-end=\"1750\">\r\n<p data-start=\"1400\" data-end=\"1427\"><strong data-start=\"1400\" data-end=\"1425\">Principal Values Only<\/strong><\/p>\r\n\r\n<ul data-start=\"1431\" data-end=\"1750\">\r\n \t<li data-start=\"1431\" data-end=\"1543\">\r\n<p data-start=\"1433\" data-end=\"1543\">[latex]\\sin^{-1}(x)[\/latex] returns an angle in [latex]\\left[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right][\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1547\" data-end=\"1634\">\r\n<p data-start=\"1549\" data-end=\"1634\">[latex]\\cos^{-1}(x)[\/latex] returns an angle in [latex]\\left[0, \\pi\\right][\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1638\" data-end=\"1750\">\r\n<p data-start=\"1640\" data-end=\"1750\">[latex]\\tan^{-1}(x)[\/latex] returns an angle in [latex]\\left(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right)[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/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-ehhahehf-4M62l7m1FJ4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4M62l7m1FJ4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ehhahehf-4M62l7m1FJ4\" 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=14661399&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ehhahehf-4M62l7m1FJ4&vembed=0&video_id=4M62l7m1FJ4&video_target=tpm-plugin-ehhahehf-4M62l7m1FJ4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Calculator+to+evaluate+inverse+trig+function+%7C+Trigonometry+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Calculator to evaluate inverse trig function | Trigonometry | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Solving Right Triangles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"366\" data-end=\"1071\">When solving right triangles, we often know two sides and want to find an angle. Inverse trigonometric functions let us \u201cwork backward\u201d from side ratios to angles. For example, if we know the opposite and adjacent sides, the ratio [latex]\\dfrac{\\text{opp}}{\\text{adj}}[\/latex] gives tangent, so we use [latex]\\tan^{-1}[\/latex] to find the angle. This process is especially useful in applications like surveying, construction, and navigation where angles must be determined from measurements of side lengths.<\/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: Solving Right Triangles with inverse Trig<\/strong>\r\n<ol>\r\n \t<li data-start=\"741\" data-end=\"998\">\r\n<p data-start=\"744\" data-end=\"775\"><strong data-start=\"744\" data-end=\"773\">Choose the Right Function<\/strong><\/p>\r\n\r\n<ul data-start=\"779\" data-end=\"998\">\r\n \t<li data-start=\"779\" data-end=\"850\">\r\n<p data-start=\"781\" data-end=\"850\">Use [latex]\\sin^{-1}[\/latex] when you know opposite and hypotenuse.<\/p>\r\n<\/li>\r\n \t<li data-start=\"854\" data-end=\"925\">\r\n<p data-start=\"856\" data-end=\"925\">Use [latex]\\cos^{-1}[\/latex] when you know adjacent and hypotenuse.<\/p>\r\n<\/li>\r\n \t<li data-start=\"929\" data-end=\"998\">\r\n<p data-start=\"931\" data-end=\"998\">Use [latex]\\tan^{-1}[\/latex] when you know opposite and adjacent.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1000\" data-end=\"1167\">\r\n<p data-start=\"1003\" data-end=\"1025\"><strong data-start=\"1003\" data-end=\"1023\">Set Up the Ratio<\/strong><\/p>\r\n<\/li>\r\n \t<li data-start=\"1000\" data-end=\"1167\">\r\n<p data-start=\"1003\" data-end=\"1025\"><strong data-start=\"1172\" data-end=\"1202\">Apply the Inverse Function<\/strong><\/p>\r\n<\/li>\r\n \t<li data-start=\"1343\" data-end=\"1623\">\r\n<p data-start=\"1346\" data-end=\"1372\"><strong data-start=\"1346\" data-end=\"1370\">Find the Other Angle<\/strong><\/p>\r\n\r\n<ul data-start=\"1376\" data-end=\"1623\">\r\n \t<li data-start=\"1376\" data-end=\"1484\">\r\n<p data-start=\"1378\" data-end=\"1484\">In a right triangle, the acute angles sum to [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1488\" data-end=\"1623\">\r\n<p data-start=\"1490\" data-end=\"1623\">If one angle is found with inverse trig, subtract from [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]) to get the other.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/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-fdedeace-LN_9BzwgjQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LN_9BzwgjQk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdedeace-LN_9BzwgjQk\" 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=14661400&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdedeace-LN_9BzwgjQk&vembed=0&video_id=LN_9BzwgjQk&video_target=tpm-plugin-fdedeace-LN_9BzwgjQk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+use+Inverse+Trig+to+Find+Missing+Angles+in+a+Right+Triangle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use Inverse Trig to Find Missing Angles in a Right Triangle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Finding Exact Values of Composite Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"403\" data-end=\"887\">Composite expressions like [latex]\\sin(\\cos^{-1} x)[\/latex] or [latex]\\tan(\\sin^{-1} x)[\/latex] ask us to evaluate one trig function of an inverse trig function. These look complicated, but the strategy is simple: treat the inner inverse as an angle, build a right triangle that matches it, and then use the triangle to find the requested function value. The restricted ranges of inverse trig functions guarantee the angle is in a specific quadrant, which tells us the correct sign.<\/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: Composite Functions with Inverse Trig<\/strong>\r\n<ol data-start=\"951\" data-end=\"2276\">\r\n \t<li data-start=\"951\" data-end=\"1094\">\r\n<p data-start=\"954\" data-end=\"999\"><strong data-start=\"954\" data-end=\"997\">Think of the Inner Function as an Angle<\/strong><\/p>\r\n\r\n<ul data-start=\"1003\" data-end=\"1094\">\r\n \t<li data-start=\"1003\" data-end=\"1094\">\r\n<p data-start=\"1005\" data-end=\"1094\">Example: Let [latex]\\theta = \\cos^{-1}(x)[\/latex]. Then [latex]\\cos \\theta = x[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1096\" data-end=\"1281\">\r\n<p data-start=\"1099\" data-end=\"1127\"><strong data-start=\"1099\" data-end=\"1125\">Build a Right Triangle<\/strong><\/p>\r\n\r\n<ul data-start=\"1131\" data-end=\"1281\">\r\n \t<li data-start=\"1131\" data-end=\"1219\">\r\n<p data-start=\"1133\" data-end=\"1219\">Use the ratio given by sine, cosine, or tangent to assign sides of a right triangle.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1223\" data-end=\"1281\">\r\n<p data-start=\"1225\" data-end=\"1281\">Fill in the missing side with the Pythagorean Theorem.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1283\" data-end=\"1441\">\r\n<p data-start=\"1286\" data-end=\"1319\"><strong data-start=\"1286\" data-end=\"1317\">Evaluate the Outer Function<\/strong><\/p>\r\n\r\n<ul data-start=\"1323\" data-end=\"1441\">\r\n \t<li data-start=\"1323\" data-end=\"1376\">\r\n<p data-start=\"1325\" data-end=\"1376\">Use the triangle to find the required trig value.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1380\" data-end=\"1441\">\r\n<p data-start=\"1382\" data-end=\"1441\">Example: [latex]\\sin(\\cos^{-1} x) = \\sqrt{1-x^2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1443\" data-end=\"1753\">\r\n<p data-start=\"1446\" data-end=\"1470\"><strong data-start=\"1446\" data-end=\"1468\">Check the Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"1474\" data-end=\"1753\">\r\n \t<li data-start=\"1474\" data-end=\"1542\">\r\n<p data-start=\"1476\" data-end=\"1542\">[latex]\\sin^{-1}(x)[\/latex] gives an angle in Quadrants I or IV.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1546\" data-end=\"1614\">\r\n<p data-start=\"1548\" data-end=\"1614\">[latex]\\cos^{-1}(x)[\/latex] gives an angle in Quadrants I or II.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1618\" data-end=\"1686\">\r\n<p data-start=\"1620\" data-end=\"1686\">[latex]\\tan^{-1}(x)[\/latex] gives an angle in Quadrants I or IV.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1690\" data-end=\"1753\">\r\n<p data-start=\"1692\" data-end=\"1753\">This determines whether the answer is positive or negative.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1755\" data-end=\"2032\">\r\n<p data-start=\"1758\" data-end=\"1790\"><strong data-start=\"1758\" data-end=\"1788\">Common Results to Remember<\/strong><\/p>\r\n\r\n<ul data-start=\"1794\" data-end=\"2032\">\r\n \t<li data-start=\"1794\" data-end=\"1845\">\r\n<p data-start=\"1796\" data-end=\"1845\">[latex]\\sin(\\cos^{-1} x) = \\sqrt{1-x^2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1849\" data-end=\"1900\">\r\n<p data-start=\"1851\" data-end=\"1900\">[latex]\\cos(\\sin^{-1} x) = \\sqrt{1-x^2}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1904\" data-end=\"1966\">\r\n<p data-start=\"1906\" data-end=\"1966\">[latex]\\tan(\\sin^{-1} x) = \\dfrac{x}{\\sqrt{1-x^2}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1970\" data-end=\"2032\">\r\n<p data-start=\"1972\" data-end=\"2032\">[latex]\\tan(\\cos^{-1} x) = \\dfrac{\\sqrt{1-x^2}}{x}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"2034\" data-end=\"2276\">\r\n<p data-start=\"2037\" data-end=\"2058\"><strong data-start=\"2037\" data-end=\"2056\">Worked Examples<\/strong><\/p>\r\n\r\n<ul data-start=\"2062\" data-end=\"2276\">\r\n \t<li data-start=\"2062\" data-end=\"2125\">\r\n<p data-start=\"2064\" data-end=\"2125\">[latex]\\sin(\\cos^{-1}(\\dfrac{3}{5})) = \\dfrac{4}{5}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"2129\" data-end=\"2194\">\r\n<p data-start=\"2131\" data-end=\"2194\">[latex]\\tan(\\sin^{-1}(\\dfrac{5}{13})) = \\dfrac{5}{12}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"2198\" data-end=\"2276\">\r\n<p data-start=\"2200\" data-end=\"2276\">[latex]\\cos(\\tan^{-1}(-\\dfrac{\\sqrt{3}}{3})) = \\dfrac{\\sqrt{3}}{2}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the exact value of [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[reveal-answer q=\"852395\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"852395\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Let [latex]\\theta = \\cos^{-1}\\left(\\frac{3}{5}\\right)[\/latex]. Then [latex]\\cos\\theta = \\frac{3}{5}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Build a right triangle where [latex]\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{3}{5}[\/latex]:<\/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\">Adjacent side = 3<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Hypotenuse = 5<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Opposite side = ?<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Use the Pythagorean Theorem: [latex] \\begin{aligned} \\text{opposite}^2 + 3^2 &amp;= 5^2 \\ \\text{opposite}^2 + 9 &amp;= 25 \\ \\text{opposite}^2 &amp;= 16 \\ \\text{opposite} &amp;= 4 \\quad \\text{(positive since } \\theta \\text{ is in Quadrant I)} \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Now find sine: [latex] \\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{4}{5} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Therefore, [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right) = \\frac{4}{5}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section>\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-fbaecbed-tMtB6-r-5Hs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tMtB6-r-5Hs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbaecbed-tMtB6-r-5Hs\" 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=14661401&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fbaecbed-tMtB6-r-5Hs&vembed=0&video_id=tMtB6-r-5Hs&video_target=tpm-plugin-fbaecbed-tMtB6-r-5Hs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluating+Compositions+of+Trig+and+Inverse+Trig+Functions+by+Hand_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Compositions of Trig and Inverse Trig Functions by Hand\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Understand the domain restrictions on inverse sine, cosine, and tangent<\/li>\n<li>Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.<\/li>\n<li>Use a calculator to evaluate inverse trigonometric functions.<\/li>\n<li>Use inverse trigonometric functions to solve right triangles.<\/li>\n<li>Find exact values of composite functions with inverse trigonometric functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Domain Restrictions on Inverse Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Inverse trigonometric functions \u201cundo\u201d sine, cosine, and tangent. But since sine, cosine, and tangent are <strong data-start=\"311\" data-end=\"323\">periodic<\/strong>, they are not one-to-one over their entire domains. To make their inverses valid functions, we must <strong data-start=\"424\" data-end=\"447\">restrict the domain<\/strong> of each trig function to an interval where it is one-to-one and covers all possible outputs. These restricted intervals are chosen so the inverse functions give a <strong data-start=\"611\" data-end=\"640\">single, consistent answer<\/strong>.<\/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: Domain Restrictions for Inverse Trig<\/strong><\/p>\n<ol>\n<li data-start=\"706\" data-end=\"944\">\n<p data-start=\"709\" data-end=\"761\"><strong data-start=\"709\" data-end=\"759\">Inverse Sine ([latex]y = \\sin^{-1}(x)[\/latex])<\/strong><\/p>\n<ul data-start=\"765\" data-end=\"944\">\n<li data-start=\"765\" data-end=\"808\">\n<p data-start=\"767\" data-end=\"808\">Domain: [latex]-1 \\leq x \\leq 1[\/latex]<\/p>\n<\/li>\n<li data-start=\"812\" data-end=\"880\">\n<p data-start=\"814\" data-end=\"880\">Range: [latex]-\\dfrac{\\pi}{2} \\leq y \\leq \\dfrac{\\pi}{2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"884\" data-end=\"944\">\n<p data-start=\"886\" data-end=\"944\">Chosen because sine is one-to-one in Quadrants I and IV.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"949\" data-end=\"1003\"><strong data-start=\"949\" data-end=\"1001\">Inverse Cosine ([latex]y = \\cos^{-1}(x)[\/latex])<\/strong>\n<ul data-start=\"1007\" data-end=\"1163\">\n<li data-start=\"1007\" data-end=\"1050\">\n<p data-start=\"1009\" data-end=\"1050\">Domain: [latex]-1 \\leq x \\leq 1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1054\" data-end=\"1097\">\n<p data-start=\"1056\" data-end=\"1097\">Range: [latex]0 \\leq y \\leq \\pi[\/latex]<\/p>\n<\/li>\n<li data-start=\"1101\" data-end=\"1163\">\n<p data-start=\"1103\" data-end=\"1163\">Chosen because cosine is one-to-one in Quadrants I and II.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1168\" data-end=\"1223\"><strong data-start=\"1168\" data-end=\"1221\">Inverse Tangent ([latex]y = \\tan^{-1}(x)[\/latex])<\/strong>\n<ul data-start=\"1227\" data-end=\"1421\">\n<li data-start=\"1227\" data-end=\"1288\">\n<p data-start=\"1229\" data-end=\"1288\">Domain: all real numbers [latex](-\\infty, \\infty)[\/latex]<\/p>\n<\/li>\n<li data-start=\"1292\" data-end=\"1354\">\n<p data-start=\"1294\" data-end=\"1354\">Range: [latex]-\\dfrac{\\pi}{2} < y < \\dfrac{\\pi}{2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1358\" data-end=\"1421\">\n<p data-start=\"1360\" data-end=\"1421\">Chosen because tangent is one-to-one in Quadrants I and IV.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hfcfhhhc-YXWKpgmLgHk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YXWKpgmLgHk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hfcfhhhc-YXWKpgmLgHk\" 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=14661398&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hfcfhhhc-YXWKpgmLgHk&#38;vembed=0&#38;video_id=YXWKpgmLgHk&#38;video_target=tpm-plugin-hfcfhhhc-YXWKpgmLgHk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Inverse+Trigonometric+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInverse Trigonometric Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Determine whether each value is in the domain of the given inverse function. If it is, find the exact value.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]\\sin^{-1}\\left(\\frac{3}{2}\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\cos^{-1}\\left(-\\frac{1}{2}\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]\\tan^{-1}(5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q420918\">Show Solution<\/button><\/p>\n<div id=\"q420918\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) The domain of [latex]\\sin^{-1}(x)[\/latex] is [latex][-1, 1][\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]\\frac{3}{2} = 1.5 > 1[\/latex], this value is not in the domain.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">[latex]\\sin^{-1}\\left(\\frac{3}{2}\\right)[\/latex] is undefined.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) The domain of [latex]\\cos^{-1}(x)[\/latex] is [latex][-1, 1][\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]-1 \\leq -\\frac{1}{2} \\leq 1[\/latex], this value is in the domain.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">We need an angle in [latex][0, \\pi][\/latex] whose cosine is [latex]-\\frac{1}{2}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\cos^{-1}\\left(-\\frac{1}{2}\\right) = \\frac{2\\pi}{3}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">(This is in Quadrant II, where cosine is negative)<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">c) The domain of [latex]\\tan^{-1}(x)[\/latex] is all real numbers [latex](-\\infty, \\infty)[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since 5 is a real number, this value is in the domain.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">[latex]\\tan^{-1}(5) \\approx 1.3734[\/latex] radians (or about [latex]78.69\u00b0[\/latex])<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<h2>Finding Exact Values of Inverse Trigonometric Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"366\" data-end=\"1071\">Inverse trig functions return a <strong data-start=\"484\" data-end=\"503\">principal angle<\/strong> whose trig value matches the input. Because sine, cosine, and tangent are periodic, their inverses must be restricted to certain ranges so they give a single, consistent answer. To find exact values, use the <strong data-start=\"712\" data-end=\"733\">special triangles<\/strong> ([latex]30^\\circ[\/latex]\u2013[latex]60^\\circ[\/latex]\u2013[latex]90^\\circ[\/latex] and [latex]45^\\circ[\/latex]\u2013[latex]45^\\circ[\/latex]\u2013[latex]90^\\circ[\/latex]) or the unit circle. For composite expressions like [latex]\\sin(\\cos^{-1} x)[\/latex] or [latex]\\tan(\\sin^{-1} x)[\/latex], build a right triangle from the inner inverse, then use it to evaluate the requested function \u2014 always checking the <strong data-start=\"1121\" data-end=\"1129\">sign<\/strong> based on the inverse\u2019s range.<\/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: Exact Value with Inverse Sine, Cosine, and Tangent<\/strong><\/p>\n<ol>\n<li data-start=\"1239\" data-end=\"1501\">\n<p data-start=\"1242\" data-end=\"1274\"><strong data-start=\"1242\" data-end=\"1272\">Know the Principal Ranges:<\/strong><\/p>\n<ul data-start=\"1278\" data-end=\"1501\">\n<li data-start=\"1278\" data-end=\"1358\">\n<p data-start=\"1280\" data-end=\"1358\">[latex]\\sin^{-1}(x) \\in \\left[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right][\/latex]<\/p>\n<\/li>\n<li data-start=\"1362\" data-end=\"1417\">\n<p data-start=\"1364\" data-end=\"1417\">[latex]\\cos^{-1}(x) \\in \\left[0, \\pi\\right][\/latex]<\/p>\n<\/li>\n<li data-start=\"1421\" data-end=\"1501\">\n<p data-start=\"1423\" data-end=\"1501\">[latex]\\tan^{-1}(x) \\in \\left(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right)[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1503\" data-end=\"1725\">\n<p data-start=\"1506\" data-end=\"1527\"><strong data-start=\"1506\" data-end=\"1525\">Special Angles:<\/strong><\/p>\n<ul data-start=\"1531\" data-end=\"1725\">\n<li data-start=\"1531\" data-end=\"1601\">\n<p data-start=\"1533\" data-end=\"1601\">[latex]\\sin^{-1}\\left(\\dfrac{1}{2}\\right)=\\dfrac{\\pi}{6}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1605\" data-end=\"1675\">\n<p data-start=\"1607\" data-end=\"1675\">[latex]\\cos^{-1}\\left(\\dfrac{1}{2}\\right)=\\dfrac{\\pi}{3}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1679\" data-end=\"1725\">\n<p data-start=\"1681\" data-end=\"1725\">[latex]\\tan^{-1}(1)=\\dfrac{\\pi}{4}[\/latex]<\/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]\">Find the exact value of each expression.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]\\sin^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]\\cos^{-1}(0)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]\\tan^{-1}(-1)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q829825\">Show Solution<\/button><\/p>\n<div id=\"q829825\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) We need an angle [latex]\\theta[\/latex] in [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex] where [latex]\\sin\\theta = \\frac{\\sqrt{3}}{2}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">From special angles: [latex]\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]\\frac{\\pi}{3}[\/latex] is in the range [latex]\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right][\/latex]:<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\sin^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right) = \\frac{\\pi}{3}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) We need an angle [latex]\\theta[\/latex] in [latex][0, \\pi][\/latex] where [latex]\\cos\\theta = 0[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">From the unit circle: [latex]\\cos\\left(\\frac{\\pi}{2}\\right) = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\cos^{-1}(0) = \\frac{\\pi}{2}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">c) We need an angle [latex]\\theta[\/latex] in [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex] where [latex]\\tan\\theta = -1[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">From special angles: [latex]\\tan\\left(-\\frac{\\pi}{4}\\right) = -1[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]-\\frac{\\pi}{4}[\/latex] is in the range [latex]\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)[\/latex]:<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\tan^{-1}(-1) = -\\frac{\\pi}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<h2>Evaluating Inverse Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"366\" data-end=\"1071\">Inverse trigonometric functions allow us to find the angle when we know the value of sine, cosine, or tangent. Since most values do not correspond to \u201cspecial angles,\u201d we rely on a calculator for decimal approximations. The calculator gives the <strong data-start=\"678\" data-end=\"697\">principal value<\/strong> of the inverse function, which comes from the restricted range of each inverse. To get correct results, the most important step is setting the calculator to the right mode: degrees if the answer should be in degrees, or radians if the answer should be in radians.<\/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 a Calculator for Inverse Trig<\/strong><\/p>\n<ol>\n<li data-start=\"1025\" data-end=\"1158\">\n<p data-start=\"1028\" data-end=\"1054\"><strong data-start=\"1028\" data-end=\"1052\">Check the Mode First<\/strong><\/p>\n<ul data-start=\"1058\" data-end=\"1158\">\n<li data-start=\"1058\" data-end=\"1106\">\n<p data-start=\"1060\" data-end=\"1106\">Degree mode if you want an angle in degrees.<\/p>\n<\/li>\n<li data-start=\"1110\" data-end=\"1158\">\n<p data-start=\"1112\" data-end=\"1158\">Radian mode if you want an angle in radians.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1160\" data-end=\"1395\">\n<p data-start=\"1163\" data-end=\"1189\"><strong data-start=\"1163\" data-end=\"1187\">Use the Inverse Keys<\/strong><\/p>\n<ul data-start=\"1193\" data-end=\"1395\">\n<li data-start=\"1193\" data-end=\"1305\">\n<p data-start=\"1195\" data-end=\"1305\">Most calculators label them as [latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], [latex]\\tan^{-1}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1309\" data-end=\"1395\">\n<p data-start=\"1311\" data-end=\"1395\">Enter the number first or press the function button first (depends on calculator).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1397\" data-end=\"1750\">\n<p data-start=\"1400\" data-end=\"1427\"><strong data-start=\"1400\" data-end=\"1425\">Principal Values Only<\/strong><\/p>\n<ul data-start=\"1431\" data-end=\"1750\">\n<li data-start=\"1431\" data-end=\"1543\">\n<p data-start=\"1433\" data-end=\"1543\">[latex]\\sin^{-1}(x)[\/latex] returns an angle in [latex]\\left[-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right][\/latex].<\/p>\n<\/li>\n<li data-start=\"1547\" data-end=\"1634\">\n<p data-start=\"1549\" data-end=\"1634\">[latex]\\cos^{-1}(x)[\/latex] returns an angle in [latex]\\left[0, \\pi\\right][\/latex].<\/p>\n<\/li>\n<li data-start=\"1638\" data-end=\"1750\">\n<p data-start=\"1640\" data-end=\"1750\">[latex]\\tan^{-1}(x)[\/latex] returns an angle in [latex]\\left(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2}\\right)[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\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-ehhahehf-4M62l7m1FJ4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4M62l7m1FJ4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ehhahehf-4M62l7m1FJ4\" 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=14661399&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ehhahehf-4M62l7m1FJ4&#38;vembed=0&#38;video_id=4M62l7m1FJ4&#38;video_target=tpm-plugin-ehhahehf-4M62l7m1FJ4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Example+-+Calculator+to+evaluate+inverse+trig+function+%7C+Trigonometry+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Calculator to evaluate inverse trig function | Trigonometry | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Solving Right Triangles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"366\" data-end=\"1071\">When solving right triangles, we often know two sides and want to find an angle. Inverse trigonometric functions let us \u201cwork backward\u201d from side ratios to angles. For example, if we know the opposite and adjacent sides, the ratio [latex]\\dfrac{\\text{opp}}{\\text{adj}}[\/latex] gives tangent, so we use [latex]\\tan^{-1}[\/latex] to find the angle. This process is especially useful in applications like surveying, construction, and navigation where angles must be determined from measurements of side lengths.<\/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 Right Triangles with inverse Trig<\/strong><\/p>\n<ol>\n<li data-start=\"741\" data-end=\"998\">\n<p data-start=\"744\" data-end=\"775\"><strong data-start=\"744\" data-end=\"773\">Choose the Right Function<\/strong><\/p>\n<ul data-start=\"779\" data-end=\"998\">\n<li data-start=\"779\" data-end=\"850\">\n<p data-start=\"781\" data-end=\"850\">Use [latex]\\sin^{-1}[\/latex] when you know opposite and hypotenuse.<\/p>\n<\/li>\n<li data-start=\"854\" data-end=\"925\">\n<p data-start=\"856\" data-end=\"925\">Use [latex]\\cos^{-1}[\/latex] when you know adjacent and hypotenuse.<\/p>\n<\/li>\n<li data-start=\"929\" data-end=\"998\">\n<p data-start=\"931\" data-end=\"998\">Use [latex]\\tan^{-1}[\/latex] when you know opposite and adjacent.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1000\" data-end=\"1167\">\n<p data-start=\"1003\" data-end=\"1025\"><strong data-start=\"1003\" data-end=\"1023\">Set Up the Ratio<\/strong><\/p>\n<\/li>\n<li data-start=\"1000\" data-end=\"1167\">\n<p data-start=\"1003\" data-end=\"1025\"><strong data-start=\"1172\" data-end=\"1202\">Apply the Inverse Function<\/strong><\/p>\n<\/li>\n<li data-start=\"1343\" data-end=\"1623\">\n<p data-start=\"1346\" data-end=\"1372\"><strong data-start=\"1346\" data-end=\"1370\">Find the Other Angle<\/strong><\/p>\n<ul data-start=\"1376\" data-end=\"1623\">\n<li data-start=\"1376\" data-end=\"1484\">\n<p data-start=\"1378\" data-end=\"1484\">In a right triangle, the acute angles sum to [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1488\" data-end=\"1623\">\n<p data-start=\"1490\" data-end=\"1623\">If one angle is found with inverse trig, subtract from [latex]90^\\circ[\/latex] (or [latex]\\dfrac{\\pi}{2}[\/latex]) to get the other.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\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-fdedeace-LN_9BzwgjQk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LN_9BzwgjQk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdedeace-LN_9BzwgjQk\" 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=14661400&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fdedeace-LN_9BzwgjQk&#38;vembed=0&#38;video_id=LN_9BzwgjQk&#38;video_target=tpm-plugin-fdedeace-LN_9BzwgjQk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+use+Inverse+Trig+to+Find+Missing+Angles+in+a+Right+Triangle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to use Inverse Trig to Find Missing Angles in a Right Triangle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Finding Exact Values of Composite Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"403\" data-end=\"887\">Composite expressions like [latex]\\sin(\\cos^{-1} x)[\/latex] or [latex]\\tan(\\sin^{-1} x)[\/latex] ask us to evaluate one trig function of an inverse trig function. These look complicated, but the strategy is simple: treat the inner inverse as an angle, build a right triangle that matches it, and then use the triangle to find the requested function value. The restricted ranges of inverse trig functions guarantee the angle is in a specific quadrant, which tells us the correct sign.<\/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: Composite Functions with Inverse Trig<\/strong><\/p>\n<ol data-start=\"951\" data-end=\"2276\">\n<li data-start=\"951\" data-end=\"1094\">\n<p data-start=\"954\" data-end=\"999\"><strong data-start=\"954\" data-end=\"997\">Think of the Inner Function as an Angle<\/strong><\/p>\n<ul data-start=\"1003\" data-end=\"1094\">\n<li data-start=\"1003\" data-end=\"1094\">\n<p data-start=\"1005\" data-end=\"1094\">Example: Let [latex]\\theta = \\cos^{-1}(x)[\/latex]. Then [latex]\\cos \\theta = x[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1096\" data-end=\"1281\">\n<p data-start=\"1099\" data-end=\"1127\"><strong data-start=\"1099\" data-end=\"1125\">Build a Right Triangle<\/strong><\/p>\n<ul data-start=\"1131\" data-end=\"1281\">\n<li data-start=\"1131\" data-end=\"1219\">\n<p data-start=\"1133\" data-end=\"1219\">Use the ratio given by sine, cosine, or tangent to assign sides of a right triangle.<\/p>\n<\/li>\n<li data-start=\"1223\" data-end=\"1281\">\n<p data-start=\"1225\" data-end=\"1281\">Fill in the missing side with the Pythagorean Theorem.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1283\" data-end=\"1441\">\n<p data-start=\"1286\" data-end=\"1319\"><strong data-start=\"1286\" data-end=\"1317\">Evaluate the Outer Function<\/strong><\/p>\n<ul data-start=\"1323\" data-end=\"1441\">\n<li data-start=\"1323\" data-end=\"1376\">\n<p data-start=\"1325\" data-end=\"1376\">Use the triangle to find the required trig value.<\/p>\n<\/li>\n<li data-start=\"1380\" data-end=\"1441\">\n<p data-start=\"1382\" data-end=\"1441\">Example: [latex]\\sin(\\cos^{-1} x) = \\sqrt{1-x^2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1443\" data-end=\"1753\">\n<p data-start=\"1446\" data-end=\"1470\"><strong data-start=\"1446\" data-end=\"1468\">Check the Quadrant<\/strong><\/p>\n<ul data-start=\"1474\" data-end=\"1753\">\n<li data-start=\"1474\" data-end=\"1542\">\n<p data-start=\"1476\" data-end=\"1542\">[latex]\\sin^{-1}(x)[\/latex] gives an angle in Quadrants I or IV.<\/p>\n<\/li>\n<li data-start=\"1546\" data-end=\"1614\">\n<p data-start=\"1548\" data-end=\"1614\">[latex]\\cos^{-1}(x)[\/latex] gives an angle in Quadrants I or II.<\/p>\n<\/li>\n<li data-start=\"1618\" data-end=\"1686\">\n<p data-start=\"1620\" data-end=\"1686\">[latex]\\tan^{-1}(x)[\/latex] gives an angle in Quadrants I or IV.<\/p>\n<\/li>\n<li data-start=\"1690\" data-end=\"1753\">\n<p data-start=\"1692\" data-end=\"1753\">This determines whether the answer is positive or negative.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1755\" data-end=\"2032\">\n<p data-start=\"1758\" data-end=\"1790\"><strong data-start=\"1758\" data-end=\"1788\">Common Results to Remember<\/strong><\/p>\n<ul data-start=\"1794\" data-end=\"2032\">\n<li data-start=\"1794\" data-end=\"1845\">\n<p data-start=\"1796\" data-end=\"1845\">[latex]\\sin(\\cos^{-1} x) = \\sqrt{1-x^2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1849\" data-end=\"1900\">\n<p data-start=\"1851\" data-end=\"1900\">[latex]\\cos(\\sin^{-1} x) = \\sqrt{1-x^2}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1904\" data-end=\"1966\">\n<p data-start=\"1906\" data-end=\"1966\">[latex]\\tan(\\sin^{-1} x) = \\dfrac{x}{\\sqrt{1-x^2}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"1970\" data-end=\"2032\">\n<p data-start=\"1972\" data-end=\"2032\">[latex]\\tan(\\cos^{-1} x) = \\dfrac{\\sqrt{1-x^2}}{x}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2034\" data-end=\"2276\">\n<p data-start=\"2037\" data-end=\"2058\"><strong data-start=\"2037\" data-end=\"2056\">Worked Examples<\/strong><\/p>\n<ul data-start=\"2062\" data-end=\"2276\">\n<li data-start=\"2062\" data-end=\"2125\">\n<p data-start=\"2064\" data-end=\"2125\">[latex]\\sin(\\cos^{-1}(\\dfrac{3}{5})) = \\dfrac{4}{5}[\/latex]<\/p>\n<\/li>\n<li data-start=\"2129\" data-end=\"2194\">\n<p data-start=\"2131\" data-end=\"2194\">[latex]\\tan(\\sin^{-1}(\\dfrac{5}{13})) = \\dfrac{5}{12}[\/latex]<\/p>\n<\/li>\n<li data-start=\"2198\" data-end=\"2276\">\n<p data-start=\"2200\" data-end=\"2276\">[latex]\\cos(\\tan^{-1}(-\\dfrac{\\sqrt{3}}{3})) = \\dfrac{\\sqrt{3}}{2}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Find the exact value of [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right)[\/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=\"q852395\">Show Solution<\/button><\/p>\n<div id=\"q852395\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Let [latex]\\theta = \\cos^{-1}\\left(\\frac{3}{5}\\right)[\/latex]. Then [latex]\\cos\\theta = \\frac{3}{5}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Build a right triangle where [latex]\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{3}{5}[\/latex]:<\/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\">Adjacent side = 3<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Hypotenuse = 5<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Opposite side = ?<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Use the Pythagorean Theorem: [latex]\\begin{aligned} \\text{opposite}^2 + 3^2 &= 5^2 \\ \\text{opposite}^2 + 9 &= 25 \\ \\text{opposite}^2 &= 16 \\ \\text{opposite} &= 4 \\quad \\text{(positive since } \\theta \\text{ is in Quadrant I)} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Now find sine: [latex]\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{4}{5}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Therefore, [latex]\\sin\\left(\\cos^{-1}\\left(\\frac{3}{5}\\right)\\right) = \\frac{4}{5}[\/latex]<\/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<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-fbaecbed-tMtB6-r-5Hs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tMtB6-r-5Hs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbaecbed-tMtB6-r-5Hs\" 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=14661401&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fbaecbed-tMtB6-r-5Hs&#38;vembed=0&#38;video_id=tMtB6-r-5Hs&#38;video_target=tpm-plugin-fbaecbed-tMtB6-r-5Hs\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluating+Compositions+of+Trig+and+Inverse+Trig+Functions+by+Hand_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluating Compositions of Trig and Inverse Trig Functions by Hand\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Inverse Trigonometric Functions\",\"author\":\"\",\"organization\":\"Professor Dave Explains\",\"url\":\"https:\/\/youtu.be\/YXWKpgmLgHk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Example: Calculator to evaluate inverse trig function | Trigonometry | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/4M62l7m1FJ4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to use Inverse Trig to Find Missing Angles in a Right Triangle\",\"author\":\"\",\"organization\":\"Math n Cheese\",\"url\":\"https:\/\/youtu.be\/LN_9BzwgjQk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Evaluating Compositions of Trig and Inverse Trig Functions by Hand\",\"author\":\"\",\"organization\":\"turksvids\",\"url\":\"https:\/\/youtu.be\/tMtB6-r-5Hs\",\"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":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Inverse Trigonometric Functions","author":"","organization":"Professor Dave Explains","url":"https:\/\/youtu.be\/YXWKpgmLgHk","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Example: Calculator to evaluate inverse trig function | Trigonometry 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