{"id":1539,"date":"2025-07-25T02:39:53","date_gmt":"2025-07-25T02:39:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1539"},"modified":"2026-03-12T06:30:27","modified_gmt":"2026-03-12T06:30:27","slug":"solving-trigonometric-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-trigonometric-equations-fresh-take\/","title":{"raw":"Solving Trigonometric Equations: Fresh Take","rendered":"Solving Trigonometric Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve equations involving a single trigonometric function.<\/li>\r\n \t<li>Solve trigonometric equations that involve factoring.<\/li>\r\n \t<li>Solve trigonometric equations using fundamental identities.<\/li>\r\n \t<li>Solve trigonometric equations with multiple angles.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Equations with a Single Trigonometric Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nEquations that involve only one trigonometric function can be solved much like basic algebraic equations\u2014first isolate the trig function, then use inverse trig or the unit circle to find solutions. Because trig functions are periodic, solutions often come in infinite families. The key is to solve for a \u201cbase\u201d angle and then add multiples of the period.\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 Single-Trig Equations<\/strong>\r\n<ol>\r\n \t<li data-start=\"495\" data-end=\"611\">\r\n<p data-start=\"498\" data-end=\"524\"><strong data-start=\"498\" data-end=\"522\">Isolate the Function<\/strong><\/p>\r\n<\/li>\r\n \t<li data-start=\"613\" data-end=\"871\">\r\n<p data-start=\"616\" data-end=\"646\"><strong data-start=\"616\" data-end=\"644\">Find the Reference Angle<\/strong><\/p>\r\n\r\n<ul data-start=\"650\" data-end=\"871\">\r\n \t<li data-start=\"650\" data-end=\"737\">\r\n<p data-start=\"652\" data-end=\"737\">Use special triangles, the unit circle, or inverse trig to get the principal angle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"873\" data-end=\"1096\">\r\n<p data-start=\"876\" data-end=\"918\"><strong data-start=\"876\" data-end=\"916\">Determine All Solutions in One Cycle<\/strong><\/p>\r\n\r\n<ul data-start=\"922\" data-end=\"1096\">\r\n \t<li data-start=\"922\" data-end=\"989\">\r\n<p data-start=\"924\" data-end=\"989\">Check which quadrants have the same sign for the trig function.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1098\" data-end=\"1378\">\r\n<p data-start=\"1101\" data-end=\"1131\"><strong data-start=\"1101\" data-end=\"1129\">Add the General Solution<\/strong><\/p>\r\n\r\n<ul data-start=\"1423\" data-end=\"1547\">\r\n \t<li data-start=\"1135\" data-end=\"1188\">\r\n<p data-start=\"1137\" data-end=\"1188\">Sine and cosine repeat every [latex]2\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1192\" data-end=\"1250\">\r\n<p data-start=\"1194\" data-end=\"1250\">Tangent repeats every [latex]\\pi[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Solve [latex]2\\sin\\theta + 1 = 0[\/latex] for [latex]0 \\leq \\theta &lt; 2\\pi[\/latex].[reveal-answer q=\"single-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"single-001\"]\r\nIsolate the trig function:[latex]\\begin{align*}\r\n2\\sin\\theta + 1 &amp;= 0 \\\\\r\n2\\sin\\theta &amp;= -1 \\\\\r\n\\sin\\theta &amp;= -\\frac{1}{2}\r\n\\end{align*}[\/latex]The reference angle is [latex]\\frac{\\pi}{6}[\/latex].Since sine is negative in Quadrants III and IV:[latex]\\begin{align*}\r\n\\theta &amp;= \\pi + \\frac{\\pi}{6} = \\frac{7\\pi}{6} &amp;&amp; \\text{Quadrant III} \\\\\r\n\\theta &amp;= 2\\pi - \\frac{\\pi}{6} = \\frac{11\\pi}{6} &amp;&amp; \\text{Quadrant IV}\r\n\\end{align*}[\/latex]The solutions are [latex]\\theta = \\frac{7\\pi}{6}, \\frac{11\\pi}{6}[\/latex].\r\n[\/hidden-answer]<\/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-bbbabegb-kEcbxiLeGTc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kEcbxiLeGTc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbbabegb-kEcbxiLeGTc\" 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=14661384&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbbabegb-kEcbxiLeGTc&vembed=0&video_id=kEcbxiLeGTc&video_target=tpm-plugin-bbbabegb-kEcbxiLeGTc'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Solving+Trigonometric+Equations+By+Finding+All+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Trigonometric Equations By Finding All Solutions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Solving Trigonometric Equations Using Factoring<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"76\" data-end=\"401\">Some trigonometric equations can be written in forms that allow factoring, much like algebraic quadratics. Once factored, each piece can be set equal to zero, and the resulting simpler trig equations are solved individually. Because trig functions are periodic, each factor can produce multiple solutions across all cycles.<\/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 by Factoring<\/strong>\r\n<ol data-start=\"448\" data-end=\"1864\">\r\n \t<li data-start=\"448\" data-end=\"587\">\r\n<p data-start=\"451\" data-end=\"481\"><strong data-start=\"451\" data-end=\"479\">Set the Equation to Zero<\/strong><\/p>\r\n\r\n<ul data-start=\"485\" data-end=\"587\">\r\n \t<li data-start=\"485\" data-end=\"523\">\r\n<p data-start=\"487\" data-end=\"523\">Rearrange so one side equals zero.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"589\" data-end=\"738\">\r\n<p data-start=\"592\" data-end=\"619\"><strong data-start=\"592\" data-end=\"617\">Factor the Expression<\/strong><\/p>\r\n\r\n<ul data-start=\"623\" data-end=\"738\">\r\n \t<li data-start=\"623\" data-end=\"684\">\r\n<p data-start=\"625\" data-end=\"684\">Factor out common terms or use quadratic-style factoring.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"912\" data-end=\"1136\">\r\n<p data-start=\"915\" data-end=\"940\"><strong data-start=\"915\" data-end=\"938\">Solve Each Equation<\/strong><\/p>\r\n\r\n<ul data-start=\"944\" data-end=\"1136\">\r\n \t<li data-start=\"944\" data-end=\"1002\">\r\n<p data-start=\"946\" data-end=\"1002\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Set each factor equal to zero:<\/span><\/p>\r\n<\/li>\r\n \t<li data-start=\"944\" data-end=\"1002\">\r\n<p data-start=\"946\" data-end=\"1002\">Use the unit circle or inverse trig to find solutions.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1138\" data-end=\"1372\">\r\n<p data-start=\"1141\" data-end=\"1172\"><strong data-start=\"1141\" data-end=\"1170\">Add the General Solutions<\/strong><\/p>\r\n\r\n<ul data-start=\"1176\" data-end=\"1372\">\r\n \t<li data-start=\"1176\" data-end=\"1372\">\r\n<p data-start=\"1178\" data-end=\"1252\">Since sine and cosine have a period of [latex]2\\pi[\/latex], add [latex]\\pm 2\\pin[\/latex] to the end of each solution.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1176\" data-end=\"1372\">Since tangent has a period of [latex]\\pi[\/latex], add [latex]\\pm \\pi[\/latex] to each unique solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1374\" data-end=\"1544\">\r\n<p data-start=\"1377\" data-end=\"1398\"><strong data-start=\"1377\" data-end=\"1396\">Check Solutions<\/strong><\/p>\r\n\r\n<ul data-start=\"1638\" data-end=\"1864\">\r\n \t<li data-start=\"1402\" data-end=\"1544\">\r\n<p data-start=\"1404\" data-end=\"1544\">Always substitute back into the original equation, especially when squaring or multiplying factors may have introduced extraneous answers.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Solve [latex]2\\cos^2\\theta - \\cos\\theta = 0[\/latex] for [latex]0 \\leq \\theta &lt; 2\\pi[\/latex].[reveal-answer q=\"factor-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"factor-001\"]\r\nFactor out the common term:[latex]\\cos\\theta(2\\cos\\theta - 1) = 0[\/latex]Set each factor equal to zero:Case 1:\r\n[latex]\\cos\\theta = 0[\/latex]\r\n[latex]\\theta = \\frac{\\pi}{2}, \\frac{3\\pi}{2}[\/latex]Case 2:\r\n[latex]2\\cos\\theta - 1 = 0[\/latex]\r\n[latex]\\begin{align*}\r\n2\\cos\\theta &amp;= 1 \\\\\r\n\\cos\\theta &amp;= \\frac{1}{2}\r\n\\end{align*}[\/latex][latex]\\\\ \\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex]All solutions: [latex]\\theta = \\frac{\\pi}{3}, \\frac{\\pi}{2}, \\frac{3\\pi}{2}, \\frac{5\\pi}{3}[\/latex]\r\n[\/hidden-answer]<\/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-ecdghhfb-iRytWq90fnY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/iRytWq90fnY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ecdghhfb-iRytWq90fnY\" 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=14661385&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ecdghhfb-iRytWq90fnY&vembed=0&video_id=iRytWq90fnY&video_target=tpm-plugin-ecdghhfb-iRytWq90fnY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Pre-Calculus+Solving+Trigonometric+Equations+Algebraically+By+Factoring_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPre-Calculus Solving Trigonometric Equations Algebraically By Factoring\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Solving Trigonometric Equations with Fundamental Identities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nSome trigonometric equations cannot be solved directly but become manageable when you apply the <strong data-start=\"184\" data-end=\"210\">fundamental identities<\/strong>. Substituting reciprocal, quotient, or Pythagorean identities can reduce an equation to one involving a single trig function. Once simplified, the equation can be solved using standard methods (isolate, find reference angle, check quadrants, and add general solutions).\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 with Identities<\/strong>\r\n<ol>\r\n \t<li data-start=\"532\" data-end=\"1068\">\r\n<p data-start=\"535\" data-end=\"577\"><strong data-start=\"535\" data-end=\"575\">Look for Opportunities to Substitute<\/strong><\/p>\r\n\r\n<ul data-start=\"581\" data-end=\"1068\">\r\n \t<li data-start=\"581\" data-end=\"723\">\r\n<p data-start=\"583\" data-end=\"723\">Replace quotients: [latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex], [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"727\" data-end=\"853\">\r\n<p data-start=\"729\" data-end=\"853\">Replace reciprocals: [latex]\\sec\\theta = \\dfrac{1}{\\cos\\theta}[\/latex], [latex]\\csc\\theta = \\dfrac{1}{\\sin\\theta}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"857\" data-end=\"1068\">\r\n<p data-start=\"859\" data-end=\"888\">Use Pythagorean identities:<\/p>\r\n\r\n<ul data-start=\"894\" data-end=\"1068\">\r\n \t<li data-start=\"894\" data-end=\"948\">\r\n<p data-start=\"896\" data-end=\"948\">[latex]\\sin^{2}\\theta + \\cos^{2}\\theta = 1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"954\" data-end=\"1008\">\r\n<p data-start=\"956\" data-end=\"1008\">[latex]1 + \\tan^{2}\\theta = \\sec^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1014\" data-end=\"1068\">\r\n<p data-start=\"1016\" data-end=\"1068\">[latex]1 + \\cot^{2}\\theta = \\csc^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1070\" data-end=\"1198\">\r\n<p data-start=\"1073\" data-end=\"1107\"><strong data-start=\"1073\" data-end=\"1105\">Convert to a Single Function<\/strong><\/p>\r\n\r\n<ul data-start=\"1111\" data-end=\"1198\">\r\n \t<li data-start=\"1111\" data-end=\"1198\">\r\n<p data-start=\"1113\" data-end=\"1198\">Try to rewrite the equation so it involves only sine, only cosine, or only tangent.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1889\" data-end=\"2013\">\r\n<p data-start=\"1892\" data-end=\"1924\"><strong data-start=\"1892\" data-end=\"1922\">Check Quadrants and Ranges<\/strong><\/p>\r\n\r\n<ul data-start=\"1928\" data-end=\"2013\">\r\n \t<li data-start=\"1928\" data-end=\"2013\">\r\n<p data-start=\"1930\" data-end=\"2013\">Make sure your answers are in the correct quadrants for the trig function\u2019s sign.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"2015\" data-end=\"2233\">\r\n<p data-start=\"2018\" data-end=\"2057\"><strong data-start=\"2018\" data-end=\"2055\">Be Alert for Extraneous Solutions<\/strong><\/p>\r\n\r\n<ul data-start=\"2061\" data-end=\"2233\">\r\n \t<li data-start=\"2061\" data-end=\"2233\">\r\n<p data-start=\"2063\" data-end=\"2233\">Identities sometimes introduce undefined values (like dividing by [latex]\\cos\\theta[\/latex] when [latex]\\cos\\theta = 0[\/latex]). Always verify in the original equation.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Solve [latex]2\\sin^2\\theta = 3\\cos\\theta[\/latex] for [latex]0 \\leq \\theta &lt; 2\\pi[\/latex].[reveal-answer q=\"identity-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"identity-001\"]\r\nUse the Pythagorean identity [latex]\\sin^2\\theta = 1 - \\cos^2\\theta[\/latex]:[latex]\\\\ \\begin{align*}\r\n2(1 - \\cos^2\\theta) &amp;= 3\\cos\\theta \\\\\r\n2 - 2\\cos^2\\theta &amp;= 3\\cos\\theta \\\\\r\n-2\\cos^2\\theta - 3\\cos\\theta + 2 &amp;= 0 \\\\\r\n2\\cos^2\\theta + 3\\cos\\theta - 2 &amp;= 0\r\n\\end{align*} \\\\[\/latex]Factor:\r\n[latex]\\\\ (2\\cos\\theta - 1)(\\cos\\theta + 2) = 0[\/latex]Case 1:\r\n[latex]2\\cos\\theta - 1 = 0[\/latex]\r\n[latex]\\cos\\theta = \\frac{1}{2}[\/latex]\r\n[latex]\\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex]Case 2:\r\n[latex]\\cos\\theta + 2 = 0[\/latex]\r\n[latex]\\cos\\theta = -2[\/latex]\r\nNo solution (cosine range is [-1, 1])[latex]\\\\[\/latex]\r\nThe solutions are [latex]\\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex].\r\n[\/hidden-answer]<\/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-agefgbdh-Qk2v7zngL48\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Qk2v7zngL48?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-agefgbdh-Qk2v7zngL48\" 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=14661386&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-agefgbdh-Qk2v7zngL48&vembed=0&video_id=Qk2v7zngL48&video_target=tpm-plugin-agefgbdh-Qk2v7zngL48'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Solving+Trig+Equations+Using+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Trig Equations Using Identities\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Solving Trigonometric Equations with Multiple Angles<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nTrigonometric equations sometimes involve multiples of the variable, like [latex]\\sin(2\\theta)[\/latex] or [latex]\\cos(3\\theta)[\/latex]. To solve these, we treat the inside of the trig function as a single variable first, solve for that angle, and then divide out the multiple to find all possible solutions. Because of periodicity, multiple angles produce more solutions within a single cycle, so it\u2019s important to capture every possibility.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Solving Multiple-Angle Equations<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li>Isolate the Function<\/li>\r\n \t<li>Let a Temporary Variable Stand In<\/li>\r\n \t<li>Solve for All Angles in One Cycle<\/li>\r\n \t<li>Return to the Original Variable<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Solve [latex]\\sin(2\\theta) = \\frac{1}{2}[\/latex] for [latex]0 \\leq \\theta &lt; 2\\pi[\/latex].[reveal-answer q=\"multiple-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"multiple-001\"]Let [latex]u = 2\\theta[\/latex], so we solve[latex]\\sin u = \\frac{1}{2}[\/latex].[latex]\\\\[\/latex]Since we need [latex]0 \\leq \\theta &lt; 2\\pi[\/latex], we need [latex]0 \\leq 2\\theta &lt; 4\\pi[\/latex].<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]\r\nFind all solutions for [latex]u[\/latex] in [latex][0, 4\\pi)[\/latex]:<\/span>[latex]\\begin{align*}\r\nu &amp;= \\frac{\\pi}{6}, \\frac{5\\pi}{6} &amp;&amp; \\text{first cycle } [0, 2\\pi) \\\\\r\nu &amp;= \\frac{\\pi}{6} + 2\\pi, \\frac{5\\pi}{6} + 2\\pi &amp;&amp; \\text{second cycle } [2\\pi, 4\\pi) \\\\\r\nu &amp;= \\frac{13\\pi}{6}, \\frac{17\\pi}{6}\r\n\\end{align*}[\/latex]Since [latex]u = 2\\theta[\/latex], divide by 2:\r\n[latex]\\theta = \\frac{\\pi}{12}, \\frac{5\\pi}{12}, \\frac{13\\pi}{12}, \\frac{17\\pi}{12}[\/latex]\r\n[\/hidden-answer]<\/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-affageac-eZPEW2hVUd0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/eZPEW2hVUd0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-affageac-eZPEW2hVUd0\" 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=14661387&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-affageac-eZPEW2hVUd0&vembed=0&video_id=eZPEW2hVUd0&video_target=tpm-plugin-affageac-eZPEW2hVUd0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Trigonometric+Equations+With+Multiple+Angles+-+Trigonometry_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Trigonometric Equations With Multiple Angles - Trigonometry\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>Solve equations involving a single trigonometric function.<\/li>\n<li>Solve trigonometric equations that involve factoring.<\/li>\n<li>Solve trigonometric equations using fundamental identities.<\/li>\n<li>Solve trigonometric equations with multiple angles.<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Equations with a Single Trigonometric Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Equations that involve only one trigonometric function can be solved much like basic algebraic equations\u2014first isolate the trig function, then use inverse trig or the unit circle to find solutions. Because trig functions are periodic, solutions often come in infinite families. The key is to solve for a \u201cbase\u201d angle and then add multiples of the period.<\/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 Single-Trig Equations<\/strong><\/p>\n<ol>\n<li data-start=\"495\" data-end=\"611\">\n<p data-start=\"498\" data-end=\"524\"><strong data-start=\"498\" data-end=\"522\">Isolate the Function<\/strong><\/p>\n<\/li>\n<li data-start=\"613\" data-end=\"871\">\n<p data-start=\"616\" data-end=\"646\"><strong data-start=\"616\" data-end=\"644\">Find the Reference Angle<\/strong><\/p>\n<ul data-start=\"650\" data-end=\"871\">\n<li data-start=\"650\" data-end=\"737\">\n<p data-start=\"652\" data-end=\"737\">Use special triangles, the unit circle, or inverse trig to get the principal angle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"873\" data-end=\"1096\">\n<p data-start=\"876\" data-end=\"918\"><strong data-start=\"876\" data-end=\"916\">Determine All Solutions in One Cycle<\/strong><\/p>\n<ul data-start=\"922\" data-end=\"1096\">\n<li data-start=\"922\" data-end=\"989\">\n<p data-start=\"924\" data-end=\"989\">Check which quadrants have the same sign for the trig function.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1098\" data-end=\"1378\">\n<p data-start=\"1101\" data-end=\"1131\"><strong data-start=\"1101\" data-end=\"1129\">Add the General Solution<\/strong><\/p>\n<ul data-start=\"1423\" data-end=\"1547\">\n<li data-start=\"1135\" data-end=\"1188\">\n<p data-start=\"1137\" data-end=\"1188\">Sine and cosine repeat every [latex]2\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"1192\" data-end=\"1250\">\n<p data-start=\"1194\" data-end=\"1250\">Tangent repeats every [latex]\\pi[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Solve [latex]2\\sin\\theta + 1 = 0[\/latex] for [latex]0 \\leq \\theta < 2\\pi[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsingle-001\">Show Solution<\/button><\/p>\n<div id=\"qsingle-001\" class=\"hidden-answer\" style=\"display: none\">\nIsolate the trig function:[latex]\\begin{align*}  2\\sin\\theta + 1 &= 0 \\\\  2\\sin\\theta &= -1 \\\\  \\sin\\theta &= -\\frac{1}{2}  \\end{align*}[\/latex]The reference angle is [latex]\\frac{\\pi}{6}[\/latex].Since sine is negative in Quadrants III and IV:[latex]\\begin{align*}  \\theta &= \\pi + \\frac{\\pi}{6} = \\frac{7\\pi}{6} && \\text{Quadrant III} \\\\  \\theta &= 2\\pi - \\frac{\\pi}{6} = \\frac{11\\pi}{6} && \\text{Quadrant IV}  \\end{align*}[\/latex]The solutions are [latex]\\theta = \\frac{7\\pi}{6}, \\frac{11\\pi}{6}[\/latex].\n<\/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-bbbabegb-kEcbxiLeGTc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kEcbxiLeGTc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbbabegb-kEcbxiLeGTc\" 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=14661384&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbbabegb-kEcbxiLeGTc&#38;vembed=0&#38;video_id=kEcbxiLeGTc&#38;video_target=tpm-plugin-bbbabegb-kEcbxiLeGTc\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Solving+Trigonometric+Equations+By+Finding+All+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Trigonometric Equations By Finding All Solutions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Solving Trigonometric Equations Using Factoring<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"76\" data-end=\"401\">Some trigonometric equations can be written in forms that allow factoring, much like algebraic quadratics. Once factored, each piece can be set equal to zero, and the resulting simpler trig equations are solved individually. Because trig functions are periodic, each factor can produce multiple solutions across all cycles.<\/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 by Factoring<\/strong><\/p>\n<ol data-start=\"448\" data-end=\"1864\">\n<li data-start=\"448\" data-end=\"587\">\n<p data-start=\"451\" data-end=\"481\"><strong data-start=\"451\" data-end=\"479\">Set the Equation to Zero<\/strong><\/p>\n<ul data-start=\"485\" data-end=\"587\">\n<li data-start=\"485\" data-end=\"523\">\n<p data-start=\"487\" data-end=\"523\">Rearrange so one side equals zero.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"589\" data-end=\"738\">\n<p data-start=\"592\" data-end=\"619\"><strong data-start=\"592\" data-end=\"617\">Factor the Expression<\/strong><\/p>\n<ul data-start=\"623\" data-end=\"738\">\n<li data-start=\"623\" data-end=\"684\">\n<p data-start=\"625\" data-end=\"684\">Factor out common terms or use quadratic-style factoring.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"912\" data-end=\"1136\">\n<p data-start=\"915\" data-end=\"940\"><strong data-start=\"915\" data-end=\"938\">Solve Each Equation<\/strong><\/p>\n<ul data-start=\"944\" data-end=\"1136\">\n<li data-start=\"944\" data-end=\"1002\">\n<p data-start=\"946\" data-end=\"1002\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Set each factor equal to zero:<\/span><\/p>\n<\/li>\n<li data-start=\"944\" data-end=\"1002\">\n<p data-start=\"946\" data-end=\"1002\">Use the unit circle or inverse trig to find solutions.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1138\" data-end=\"1372\">\n<p data-start=\"1141\" data-end=\"1172\"><strong data-start=\"1141\" data-end=\"1170\">Add the General Solutions<\/strong><\/p>\n<ul data-start=\"1176\" data-end=\"1372\">\n<li data-start=\"1176\" data-end=\"1372\">\n<p data-start=\"1178\" data-end=\"1252\">Since sine and cosine have a period of [latex]2\\pi[\/latex], add [latex]\\pm 2\\pin[\/latex] to the end of each solution.<\/p>\n<\/li>\n<li data-start=\"1176\" data-end=\"1372\">Since tangent has a period of [latex]\\pi[\/latex], add [latex]\\pm \\pi[\/latex] to each unique solution.<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1374\" data-end=\"1544\">\n<p data-start=\"1377\" data-end=\"1398\"><strong data-start=\"1377\" data-end=\"1396\">Check Solutions<\/strong><\/p>\n<ul data-start=\"1638\" data-end=\"1864\">\n<li data-start=\"1402\" data-end=\"1544\">\n<p data-start=\"1404\" data-end=\"1544\">Always substitute back into the original equation, especially when squaring or multiplying factors may have introduced extraneous answers.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Solve [latex]2\\cos^2\\theta - \\cos\\theta = 0[\/latex] for [latex]0 \\leq \\theta < 2\\pi[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfactor-001\">Show Solution<\/button><\/p>\n<div id=\"qfactor-001\" class=\"hidden-answer\" style=\"display: none\">\nFactor out the common term:[latex]\\cos\\theta(2\\cos\\theta - 1) = 0[\/latex]Set each factor equal to zero:Case 1:<br \/>\n[latex]\\cos\\theta = 0[\/latex]<br \/>\n[latex]\\theta = \\frac{\\pi}{2}, \\frac{3\\pi}{2}[\/latex]Case 2:<br \/>\n[latex]2\\cos\\theta - 1 = 0[\/latex]<br \/>\n[latex]\\begin{align*}  2\\cos\\theta &= 1 \\\\  \\cos\\theta &= \\frac{1}{2}  \\end{align*}[\/latex][latex]\\\\ \\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex]All solutions: [latex]\\theta = \\frac{\\pi}{3}, \\frac{\\pi}{2}, \\frac{3\\pi}{2}, \\frac{5\\pi}{3}[\/latex]\n<\/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-ecdghhfb-iRytWq90fnY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/iRytWq90fnY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ecdghhfb-iRytWq90fnY\" 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=14661385&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ecdghhfb-iRytWq90fnY&#38;vembed=0&#38;video_id=iRytWq90fnY&#38;video_target=tpm-plugin-ecdghhfb-iRytWq90fnY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Pre-Calculus+Solving+Trigonometric+Equations+Algebraically+By+Factoring_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPre-Calculus Solving Trigonometric Equations Algebraically By Factoring\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Solving Trigonometric Equations with Fundamental Identities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Some trigonometric equations cannot be solved directly but become manageable when you apply the <strong data-start=\"184\" data-end=\"210\">fundamental identities<\/strong>. Substituting reciprocal, quotient, or Pythagorean identities can reduce an equation to one involving a single trig function. Once simplified, the equation can be solved using standard methods (isolate, find reference angle, check quadrants, and add general solutions).<\/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 with Identities<\/strong><\/p>\n<ol>\n<li data-start=\"532\" data-end=\"1068\">\n<p data-start=\"535\" data-end=\"577\"><strong data-start=\"535\" data-end=\"575\">Look for Opportunities to Substitute<\/strong><\/p>\n<ul data-start=\"581\" data-end=\"1068\">\n<li data-start=\"581\" data-end=\"723\">\n<p data-start=\"583\" data-end=\"723\">Replace quotients: [latex]\\tan\\theta = \\dfrac{\\sin\\theta}{\\cos\\theta}[\/latex], [latex]\\cot\\theta = \\dfrac{\\cos\\theta}{\\sin\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"727\" data-end=\"853\">\n<p data-start=\"729\" data-end=\"853\">Replace reciprocals: [latex]\\sec\\theta = \\dfrac{1}{\\cos\\theta}[\/latex], [latex]\\csc\\theta = \\dfrac{1}{\\sin\\theta}[\/latex].<\/p>\n<\/li>\n<li data-start=\"857\" data-end=\"1068\">\n<p data-start=\"859\" data-end=\"888\">Use Pythagorean identities:<\/p>\n<ul data-start=\"894\" data-end=\"1068\">\n<li data-start=\"894\" data-end=\"948\">\n<p data-start=\"896\" data-end=\"948\">[latex]\\sin^{2}\\theta + \\cos^{2}\\theta = 1[\/latex]<\/p>\n<\/li>\n<li data-start=\"954\" data-end=\"1008\">\n<p data-start=\"956\" data-end=\"1008\">[latex]1 + \\tan^{2}\\theta = \\sec^{2}\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"1014\" data-end=\"1068\">\n<p data-start=\"1016\" data-end=\"1068\">[latex]1 + \\cot^{2}\\theta = \\csc^{2}\\theta[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1070\" data-end=\"1198\">\n<p data-start=\"1073\" data-end=\"1107\"><strong data-start=\"1073\" data-end=\"1105\">Convert to a Single Function<\/strong><\/p>\n<ul data-start=\"1111\" data-end=\"1198\">\n<li data-start=\"1111\" data-end=\"1198\">\n<p data-start=\"1113\" data-end=\"1198\">Try to rewrite the equation so it involves only sine, only cosine, or only tangent.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1889\" data-end=\"2013\">\n<p data-start=\"1892\" data-end=\"1924\"><strong data-start=\"1892\" data-end=\"1922\">Check Quadrants and Ranges<\/strong><\/p>\n<ul data-start=\"1928\" data-end=\"2013\">\n<li data-start=\"1928\" data-end=\"2013\">\n<p data-start=\"1930\" data-end=\"2013\">Make sure your answers are in the correct quadrants for the trig function\u2019s sign.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"2015\" data-end=\"2233\">\n<p data-start=\"2018\" data-end=\"2057\"><strong data-start=\"2018\" data-end=\"2055\">Be Alert for Extraneous Solutions<\/strong><\/p>\n<ul data-start=\"2061\" data-end=\"2233\">\n<li data-start=\"2061\" data-end=\"2233\">\n<p data-start=\"2063\" data-end=\"2233\">Identities sometimes introduce undefined values (like dividing by [latex]\\cos\\theta[\/latex] when [latex]\\cos\\theta = 0[\/latex]). Always verify in the original equation.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Solve [latex]2\\sin^2\\theta = 3\\cos\\theta[\/latex] for [latex]0 \\leq \\theta < 2\\pi[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qidentity-001\">Show Solution<\/button><\/p>\n<div id=\"qidentity-001\" class=\"hidden-answer\" style=\"display: none\">\nUse the Pythagorean identity [latex]\\sin^2\\theta = 1 - \\cos^2\\theta[\/latex]:[latex]\\\\ \\begin{align*}  2(1 - \\cos^2\\theta) &= 3\\cos\\theta \\\\  2 - 2\\cos^2\\theta &= 3\\cos\\theta \\\\  -2\\cos^2\\theta - 3\\cos\\theta + 2 &= 0 \\\\  2\\cos^2\\theta + 3\\cos\\theta - 2 &= 0  \\end{align*} \\\\[\/latex]Factor:<br \/>\n[latex]\\\\ (2\\cos\\theta - 1)(\\cos\\theta + 2) = 0[\/latex]Case 1:<br \/>\n[latex]2\\cos\\theta - 1 = 0[\/latex]<br \/>\n[latex]\\cos\\theta = \\frac{1}{2}[\/latex]<br \/>\n[latex]\\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex]Case 2:<br \/>\n[latex]\\cos\\theta + 2 = 0[\/latex]<br \/>\n[latex]\\cos\\theta = -2[\/latex]<br \/>\nNo solution (cosine range is [-1, 1])[latex]\\\\[\/latex]<br \/>\nThe solutions are [latex]\\theta = \\frac{\\pi}{3}, \\frac{5\\pi}{3}[\/latex].\n<\/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-agefgbdh-Qk2v7zngL48\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Qk2v7zngL48?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-agefgbdh-Qk2v7zngL48\" 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=14661386&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-agefgbdh-Qk2v7zngL48&#38;vembed=0&#38;video_id=Qk2v7zngL48&#38;video_target=tpm-plugin-agefgbdh-Qk2v7zngL48\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Solving+Trig+Equations+Using+Identities_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving Trig Equations Using Identities\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Solving Trigonometric Equations with Multiple Angles<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Trigonometric equations sometimes involve multiples of the variable, like [latex]\\sin(2\\theta)[\/latex] or [latex]\\cos(3\\theta)[\/latex]. To solve these, we treat the inside of the trig function as a single variable first, solve for that angle, and then divide out the multiple to find all possible solutions. Because of periodicity, multiple angles produce more solutions within a single cycle, so it\u2019s important to capture every possibility.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Solving Multiple-Angle Equations<\/strong><\/p>\n<ol>\n<li>Isolate the Function<\/li>\n<li>Let a Temporary Variable Stand In<\/li>\n<li>Solve for All Angles in One Cycle<\/li>\n<li>Return to the Original Variable<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Solve [latex]\\sin(2\\theta) = \\frac{1}{2}[\/latex] for [latex]0 \\leq \\theta < 2\\pi[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qmultiple-001\">Show Solution<\/button><\/p>\n<div id=\"qmultiple-001\" class=\"hidden-answer\" style=\"display: none\">Let [latex]u = 2\\theta[\/latex], so we solve[latex]\\sin u = \\frac{1}{2}[\/latex].[latex]\\\\[\/latex]Since we need [latex]0 \\leq \\theta < 2\\pi[\/latex], we need [latex]0 \\leq 2\\theta < 4\\pi[\/latex].<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\\\[\/latex]<br \/>\nFind all solutions for [latex]u[\/latex] in [latex][0, 4\\pi)[\/latex]:<\/span>[latex]\\begin{align*}  u &= \\frac{\\pi}{6}, \\frac{5\\pi}{6} && \\text{first cycle } [0, 2\\pi) \\\\  u &= \\frac{\\pi}{6} + 2\\pi, \\frac{5\\pi}{6} + 2\\pi && \\text{second cycle } [2\\pi, 4\\pi) \\\\  u &= \\frac{13\\pi}{6}, \\frac{17\\pi}{6}  \\end{align*}[\/latex]Since [latex]u = 2\\theta[\/latex], divide by 2:<br \/>\n[latex]\\theta = \\frac{\\pi}{12}, \\frac{5\\pi}{12}, \\frac{13\\pi}{12}, \\frac{17\\pi}{12}[\/latex]\n<\/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-affageac-eZPEW2hVUd0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/eZPEW2hVUd0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-affageac-eZPEW2hVUd0\" 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=14661387&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-affageac-eZPEW2hVUd0&#38;vembed=0&#38;video_id=eZPEW2hVUd0&#38;video_target=tpm-plugin-affageac-eZPEW2hVUd0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Trigonometric+Equations+With+Multiple+Angles+-+Trigonometry_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Trigonometric Equations With Multiple Angles &#8211; Trigonometry\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":32,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Solving Trigonometric Equations By Finding All Solutions\",\"author\":\"\",\"organization\":\"The Organic Chemistry Tutor\",\"url\":\"https:\/\/youtu.be\/kEcbxiLeGTc\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Pre-Calculus Solving Trigonometric Equations Algebraically By Factoring\",\"author\":\"\",\"organization\":\"Mr. Hernandez Teaches\",\"url\":\"https:\/\/youtu.be\/iRytWq90fnY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Solving Trig Equations Using Identities\",\"author\":\"\",\"organization\":\"Math And Physics Tutor\",\"url\":\"https:\/\/youtu.be\/Qk2v7zngL48\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How To Solve Trigonometric Equations With Multiple Angles - 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