{"id":1536,"date":"2025-07-25T02:38:59","date_gmt":"2025-07-25T02:38:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1536"},"modified":"2026-03-12T06:06:59","modified_gmt":"2026-03-12T06:06:59","slug":"double-angle-half-angle-and-reduction-formulas-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/double-angle-half-angle-and-reduction-formulas-fresh-take\/","title":{"raw":"Double Angle, Half Angle, and Reduction Formulas: Fresh Take","rendered":"Double Angle, Half Angle, and Reduction Formulas: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use double-angle formulas to find exact values.<\/li>\r\n \t<li>Use double-angle formulas to verify identities.<\/li>\r\n \t<li>Use reduction formulas to simplify an expression.<\/li>\r\n \t<li>Use half-angle formulas to find exact values.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Double-Angle Formulas for Exact Values<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"67\" data-end=\"431\">Double-angle formulas let us evaluate trig functions of angles like [latex]2\\theta[\/latex] by rewriting them in terms of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]\\sin(2 \\cdot 15^\\circ)[\/latex] or [latex]\\cos(2 \\cdot 22.5^\\circ)[\/latex].<\/p>\r\n<p data-start=\"433\" data-end=\"452\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"454\" data-end=\"696\">\r\n \t<li data-start=\"454\" data-end=\"510\">\r\n<p data-start=\"456\" data-end=\"510\">[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"511\" data-end=\"621\">\r\n<p data-start=\"513\" data-end=\"621\">[latex]\\cos(2\\theta) = \\cos^{2}\\theta - \\sin^{2}\\theta = 2\\cos^{2}\\theta - 1 = 1 - 2\\sin^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"622\" data-end=\"696\">\r\n<p data-start=\"624\" data-end=\"696\">[latex]\\tan(2\\theta) = \\dfrac{2\\tan\\theta}{1 - \\tan^{2}\\theta}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Double-Angle Formulas<\/strong>\r\n<ol>\r\n \t<li data-start=\"750\" data-end=\"932\">\r\n<p data-start=\"753\" data-end=\"791\"><strong data-start=\"753\" data-end=\"789\">Pick the Right Version of Cosine<\/strong><\/p>\r\n\r\n<ul data-start=\"795\" data-end=\"932\">\r\n \t<li data-start=\"795\" data-end=\"867\">\r\n<p data-start=\"797\" data-end=\"867\">Use [latex]2\\cos^{2}\\theta - 1[\/latex] if cosine is easy to compute.<\/p>\r\n<\/li>\r\n \t<li data-start=\"871\" data-end=\"932\">\r\n<p data-start=\"873\" data-end=\"932\">Use [latex]1 - 2\\sin^{2}\\theta[\/latex] if sine is easier.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"934\" data-end=\"1265\">\r\n<p data-start=\"937\" data-end=\"978\"><strong data-start=\"937\" data-end=\"976\">Work an Example with Special Angles<\/strong><\/p>\r\n\r\n<ul data-start=\"982\" data-end=\"1265\">\r\n \t<li data-start=\"982\" data-end=\"1040\">\r\n<p data-start=\"984\" data-end=\"1040\">Find [latex]\\sin(30^\\circ)[\/latex] using double-angle:<\/p>\r\n<\/li>\r\n \t<li data-start=\"1044\" data-end=\"1155\">\r\n<p data-start=\"1046\" data-end=\"1155\">[latex]\\sin(30^\\circ) = \\sin(2 \\cdot 15^\\circ)[\/latex]<br data-start=\"1100\" data-end=\"1103\" \/>= [latex]2\\sin(15^\\circ)\\cos(15^\\circ)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1159\" data-end=\"1265\">\r\n<p data-start=\"1161\" data-end=\"1265\">Use known exact values of [latex]\\sin(15^\\circ)[\/latex] and [latex]\\cos(15^\\circ)[\/latex] to evaluate.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1267\" data-end=\"1417\">\r\n<p data-start=\"1270\" data-end=\"1297\"><strong data-start=\"1270\" data-end=\"1295\">Simplify Step by Step<\/strong><\/p>\r\n\r\n<ul data-start=\"1301\" data-end=\"1417\">\r\n \t<li data-start=\"1301\" data-end=\"1348\">\r\n<p data-start=\"1303\" data-end=\"1348\">Always reduce radicals and fractions fully.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1352\" data-end=\"1417\">\r\n<p data-start=\"1354\" data-end=\"1417\">Keep answers exact in the form of fractions and square roots.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1419\" data-end=\"1613\">\r\n<p data-start=\"1422\" data-end=\"1446\"><strong data-start=\"1422\" data-end=\"1444\">Check the Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"1450\" data-end=\"1613\">\r\n \t<li data-start=\"1450\" data-end=\"1613\">\r\n<p data-start=\"1452\" data-end=\"1613\">The sign of [latex]\\sin(2\\theta)[\/latex], [latex]\\cos(2\\theta)[\/latex], or [latex]\\tan(2\\theta)[\/latex] depends on which quadrant [latex]2\\theta[\/latex] is in.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1615\" data-end=\"1896\">\r\n<p data-start=\"1618\" data-end=\"1682\"><strong data-start=\"1618\" data-end=\"1629\">Example: <\/strong>Find [latex]\\cos(2 \\cdot 22.5^\\circ)[\/latex].<\/p>\r\n\r\n<ul data-start=\"1686\" data-end=\"1896\">\r\n \t<li data-start=\"1686\" data-end=\"1766\">\r\n<p data-start=\"1688\" data-end=\"1766\">[latex]\\cos(45^\\circ) = \\cos^{2}(22.5^\\circ) - \\sin^{2}(22.5^\\circ)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1770\" data-end=\"1896\">\r\n<p data-start=\"1772\" data-end=\"1896\">Since [latex]\\cos(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex], this connects the exact values of half-angles to double-angles.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">If [latex]\\sin\\theta = \\frac{3}{5}[\/latex] and [latex]\\theta[\/latex] is in Quadrant II, find [latex]\\sin(2\\theta)[\/latex].[reveal-answer q=\"double-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"double-001\"]\r\nFirst, find [latex]\\cos\\theta[\/latex] using the Pythagorean identity:\r\n[latex]\\begin{align*}\r\n\\cos^2\\theta &amp;= 1 - \\sin^2\\theta \\\\\r\n&amp;= 1 - \\left(\\frac{3}{5}\\right)^2 \\\\\r\n&amp;= 1 - \\frac{9}{25} \\\\\r\n&amp;= \\frac{16}{25}\r\n\\end{align*}[\/latex]Since [latex]\\theta[\/latex] is in Quadrant II, [latex]\\cos\\theta = -\\frac{4}{5}[\/latex]Use the double-angle formula:\r\n[latex]\\begin{align*}\r\n\\sin(2\\theta) &amp;= 2\\sin\\theta\\cos\\theta \\\\\r\n&amp;= 2\\left(\\frac{3}{5}\\right)\\left(-\\frac{4}{5}\\right) \\\\\r\n&amp;= -\\frac{24}{25}\r\n\\end{align*}[\/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-cfdbdbcc-SE5SBTgrwH8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SE5SBTgrwH8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfdbdbcc-SE5SBTgrwH8\" 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=14661367&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfdbdbcc-SE5SBTgrwH8&vembed=0&video_id=SE5SBTgrwH8&video_target=tpm-plugin-cfdbdbcc-SE5SBTgrwH8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Double+Angle+Identities+%26+Formulas+of+Sin%2C+Cos+%26+Tan+-+Trigonometry_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDouble Angle Identities &amp; Formulas of Sin, Cos &amp; Tan - Trigonometry\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Verifying Identities with Double-Angle Formulas<\/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=\"469\">Double-angle formulas connect trigonometric functions of [latex]2\\theta[\/latex] to those of [latex]\\theta[\/latex]. They are powerful tools for proving that two trig expressions are equal. In verification problems, you expand one side using a double-angle identity, then simplify until it matches the other side. This shows that the identity holds for all values where both sides are defined.<\/p>\r\n<p data-start=\"471\" data-end=\"490\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"492\" data-end=\"734\">\r\n \t<li data-start=\"492\" data-end=\"548\">\r\n<p data-start=\"494\" data-end=\"548\">[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"549\" data-end=\"659\">\r\n<p data-start=\"551\" data-end=\"659\">[latex]\\cos(2\\theta) = \\cos^{2}\\theta - \\sin^{2}\\theta = 2\\cos^{2}\\theta - 1 = 1 - 2\\sin^{2}\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"660\" data-end=\"734\">\r\n<p data-start=\"662\" data-end=\"734\">[latex]\\tan(2\\theta) = \\dfrac{2\\tan\\theta}{1 - \\tan^{2}\\theta}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Double-Angle Formulas in Verifications<\/strong>\r\n<ol>\r\n \t<li data-start=\"805\" data-end=\"1123\">\r\n<p data-start=\"808\" data-end=\"839\"><strong data-start=\"808\" data-end=\"837\">Choose the Right Identity<\/strong><\/p>\r\n\r\n<ul data-start=\"843\" data-end=\"1123\">\r\n \t<li data-start=\"843\" data-end=\"944\">\r\n<p data-start=\"845\" data-end=\"944\">If the expression involves [latex]\\sin\\theta\\cos\\theta[\/latex], use [latex]\\sin(2\\theta)[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"948\" data-end=\"1062\">\r\n<p data-start=\"950\" data-end=\"1062\">If it involves [latex]\\cos^{2}\\theta[\/latex] or [latex]\\sin^{2}\\theta[\/latex], use one of the cosine formulas.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1066\" data-end=\"1123\">\r\n<p data-start=\"1068\" data-end=\"1123\">If tangent appears, use [latex]\\tan(2\\theta)[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1125\" data-end=\"1274\">\r\n<p data-start=\"1128\" data-end=\"1164\"><strong data-start=\"1128\" data-end=\"1162\">Work From the Complicated Side<\/strong><\/p>\r\n\r\n<ul data-start=\"1168\" data-end=\"1274\">\r\n \t<li data-start=\"1168\" data-end=\"1227\">\r\n<p data-start=\"1170\" data-end=\"1227\">Expand or replace terms on the more complex side first.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1231\" data-end=\"1274\">\r\n<p data-start=\"1233\" data-end=\"1274\">Leave the simpler side as the \u201ctarget.\u201d<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1276\" data-end=\"1481\">\r\n<p data-start=\"1279\" data-end=\"1306\"><strong data-start=\"1279\" data-end=\"1304\">Simplify Step by Step<\/strong><\/p>\r\n\r\n<ul data-start=\"1310\" data-end=\"1481\">\r\n \t<li data-start=\"1310\" data-end=\"1427\">\r\n<p data-start=\"1312\" data-end=\"1427\">Apply Pythagorean identities to replace [latex]\\sin^{2}\\theta[\/latex] or [latex]\\cos^{2}\\theta[\/latex] if needed.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1431\" data-end=\"1481\">\r\n<p data-start=\"1433\" data-end=\"1481\">Combine fractions or cancel factors carefully.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Verify the identity: [latex]\\frac{\\sin(2\\theta)}{1 + \\cos(2\\theta)} = \\tan\\theta[\/latex][reveal-answer q=\"double-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"double-002\"]\r\nWork with the left side:\r\n[latex]\\begin{align*}\r\n\\frac{\\sin(2\\theta)}{1 + \\cos(2\\theta)} &amp;= \\frac{2\\sin\\theta\\cos\\theta}{1 + (2\\cos^2\\theta - 1)} &amp;&amp; \\text{use double-angle formulas} \\\\\r\n&amp;= \\frac{2\\sin\\theta\\cos\\theta}{2\\cos^2\\theta} &amp;&amp; \\text{simplify denominator} \\\\\r\n&amp;= \\frac{\\sin\\theta}{\\cos\\theta} &amp;&amp; \\text{cancel } 2\\cos\\theta \\\\\r\n&amp;= \\tan\\theta &amp;&amp; \\text{quotient identity}\r\n\\end{align*}[\/latex]The identity is verified.\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-cacbbbfg-sZ1GjTqgR0s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sZ1GjTqgR0s?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cacbbbfg-sZ1GjTqgR0s\" 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=14661368&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cacbbbfg-sZ1GjTqgR0s&vembed=0&video_id=sZ1GjTqgR0s&video_target=tpm-plugin-cacbbbfg-sZ1GjTqgR0s'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+verify+an+identity+using+double+angle+formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to verify an identity using double angle formulas\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Simplifying Expressions with Reduction Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nReduction formulas (also called power-reducing formulas) allow us to rewrite even powers of sine or cosine in terms of the first power of cosine. These formulas are derived from the double-angle formulas and are especially important in calculus.\r\n\r\nThe reduction formulas are:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sin^2\\theta = \\frac{1 - \\cos(2\\theta)}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\cos^2\\theta = \\frac{1 + \\cos(2\\theta)}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan^2\\theta = \\frac{1 - \\cos(2\\theta)}{1 + \\cos(2\\theta)}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Reduction Formulas<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Apply the formula to reduce the power<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If higher powers remain, apply the formula again<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify using algebra<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div><section class=\"textbox example\">Write an equivalent expression for [latex]\\sin^4 x[\/latex] that does not involve any powers of sine or cosine greater than 1. [reveal-answer q=\"reduction-001\"]Show Solution[\/reveal-answer] [hidden-answer a=\"reduction-001\"] Apply the reduction formula twice: [latex]\\begin{align*} \\sin^4 x &amp;= (\\sin^2 x)^2 \\\\ &amp;= \\left(\\frac{1 - \\cos(2x)}{2}\\right)^2 &amp;&amp; \\text{substitute reduction formula} \\\\ &amp;= \\frac{1}{4}(1 - 2\\cos(2x) + \\cos^2(2x)) &amp;&amp; \\text{expand} \\\\ &amp;= \\frac{1}{4} - \\frac{1}{2}\\cos(2x) + \\frac{1}{4}\\left(\\frac{1 + \\cos(4x)}{2}\\right) &amp;&amp; \\text{apply formula again} \\\\ &amp;= \\frac{1}{4} - \\frac{1}{2}\\cos(2x) + \\frac{1}{8} + \\frac{1}{8}\\cos(4x) \\\\ &amp;= \\frac{3}{8} - \\frac{1}{2}\\cos(2x) + \\frac{1}{8}\\cos(4x) \\end{align*}[\/latex] [\/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-dhageega-5Ipor4q0Jd8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5Ipor4q0Jd8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dhageega-5Ipor4q0Jd8\" 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=14661369&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dhageega-5Ipor4q0Jd8&vembed=0&video_id=5Ipor4q0Jd8&video_target=tpm-plugin-dhageega-5Ipor4q0Jd8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Power+Reducing+Formulas+for+Sine+and+Cosine%2C+Example+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPower Reducing Formulas for Sine and Cosine, Example 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Half-Angle Formulas for Exact Values<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"65\" data-end=\"420\">Half-angle formulas allow us to evaluate trig functions of angles like [latex]\\dfrac{\\theta}{2}[\/latex] by rewriting them in terms of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]22.5^\\circ[\/latex] or [latex]\\dfrac{\\pi}{8}[\/latex].<\/p>\r\n<p data-start=\"422\" data-end=\"441\">The formulas are:<\/p>\r\n\r\n<ul data-start=\"443\" data-end=\"746\">\r\n \t<li data-start=\"443\" data-end=\"532\">\r\n<p data-start=\"445\" data-end=\"532\">[latex]\\sin\\left(\\dfrac{\\theta}{2}\\right) = \\pm\\sqrt{\\dfrac{1-\\cos\\theta}{2}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"533\" data-end=\"622\">\r\n<p data-start=\"535\" data-end=\"622\">[latex]\\cos\\left(\\dfrac{\\theta}{2}\\right) = \\pm\\sqrt{\\dfrac{1+\\cos\\theta}{2}}[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"623\" data-end=\"746\">\r\n<p data-start=\"625\" data-end=\"746\">[latex]\\tan\\left(\\dfrac{\\theta}{2}\\right) = \\dfrac{\\sin\\theta}{1+\\cos\\theta} = \\dfrac{1-\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"748\" data-end=\"842\">The sign (positive or negative) depends on the quadrant of [latex]\\dfrac{\\theta}{2}[\/latex].<\/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 Half-Angle Formulas<\/strong>\r\n<ol>\r\n \t<li data-start=\"894\" data-end=\"1069\">\r\n<p data-start=\"897\" data-end=\"921\"><strong data-start=\"897\" data-end=\"919\">Check the Quadrant<\/strong><\/p>\r\n\r\n<ul data-start=\"925\" data-end=\"1069\">\r\n \t<li data-start=\"925\" data-end=\"1010\">\r\n<p data-start=\"927\" data-end=\"1010\">Decide whether [latex]\\dfrac{\\theta}{2}[\/latex] is in Quadrant I, II, III, or IV.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1014\" data-end=\"1069\">\r\n<p data-start=\"1016\" data-end=\"1069\">Use this to assign the correct sign in the formula.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1071\" data-end=\"1235\">\r\n<p data-start=\"1074\" data-end=\"1113\"><strong data-start=\"1074\" data-end=\"1111\">Choose the Formula That Fits Best<\/strong><\/p>\r\n\r\n<ul data-start=\"1117\" data-end=\"1235\">\r\n \t<li data-start=\"1117\" data-end=\"1168\">\r\n<p data-start=\"1119\" data-end=\"1168\">For sine and cosine, use the square root forms.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1172\" data-end=\"1235\">\r\n<p data-start=\"1174\" data-end=\"1235\">For tangent, pick the version that avoids dividing by zero.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">Find the exact value of [latex]\\cos(15\u00b0)[\/latex] using a half-angle formula.[reveal-answer q=\"half-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"half-001\"]\r\nWrite [latex]15\u00b0 = \\frac{30\u00b0}{2}[\/latex]Since [latex]15\u00b0[\/latex] is in Quadrant I, cosine is positive.Use the half-angle formula:\r\n[latex]\\begin{align*}\r\n\\cos(15\u00b0) &amp;= \\cos\\left(\\frac{30\u00b0}{2}\\right) \\\\\r\n&amp;= +\\sqrt{\\frac{1 + \\cos(30\u00b0)}{2}} \\\\\r\n&amp;= \\sqrt{\\frac{1 + \\frac{\\sqrt{3}}{2}}{2}} \\\\\r\n&amp;= \\sqrt{\\frac{\\frac{2 + \\sqrt{3}}{2}}{2}} \\\\\r\n&amp;= \\sqrt{\\frac{2 + \\sqrt{3}}{4}} \\\\\r\n&amp;= \\frac{\\sqrt{2 + \\sqrt{3}}}{2}\r\n\\end{align*}[\/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-fadhehdg-ZncEctA2Fug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZncEctA2Fug?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fadhehdg-ZncEctA2Fug\" 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=14661370&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fadhehdg-ZncEctA2Fug&vembed=0&video_id=ZncEctA2Fug&video_target=tpm-plugin-fadhehdg-ZncEctA2Fug'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluate+the+half+angle+in+radians+for+sine_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate the half angle in radians for sine\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>Use double-angle formulas to find exact values.<\/li>\n<li>Use double-angle formulas to verify identities.<\/li>\n<li>Use reduction formulas to simplify an expression.<\/li>\n<li>Use half-angle formulas to find exact values.<\/li>\n<\/ul>\n<\/section>\n<h2>Double-Angle Formulas for Exact Values<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"67\" data-end=\"431\">Double-angle formulas let us evaluate trig functions of angles like [latex]2\\theta[\/latex] by rewriting them in terms of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]\\sin(2 \\cdot 15^\\circ)[\/latex] or [latex]\\cos(2 \\cdot 22.5^\\circ)[\/latex].<\/p>\n<p data-start=\"433\" data-end=\"452\">The formulas are:<\/p>\n<ul data-start=\"454\" data-end=\"696\">\n<li data-start=\"454\" data-end=\"510\">\n<p data-start=\"456\" data-end=\"510\">[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"511\" data-end=\"621\">\n<p data-start=\"513\" data-end=\"621\">[latex]\\cos(2\\theta) = \\cos^{2}\\theta - \\sin^{2}\\theta = 2\\cos^{2}\\theta - 1 = 1 - 2\\sin^{2}\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"622\" data-end=\"696\">\n<p data-start=\"624\" data-end=\"696\">[latex]\\tan(2\\theta) = \\dfrac{2\\tan\\theta}{1 - \\tan^{2}\\theta}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Double-Angle Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"750\" data-end=\"932\">\n<p data-start=\"753\" data-end=\"791\"><strong data-start=\"753\" data-end=\"789\">Pick the Right Version of Cosine<\/strong><\/p>\n<ul data-start=\"795\" data-end=\"932\">\n<li data-start=\"795\" data-end=\"867\">\n<p data-start=\"797\" data-end=\"867\">Use [latex]2\\cos^{2}\\theta - 1[\/latex] if cosine is easy to compute.<\/p>\n<\/li>\n<li data-start=\"871\" data-end=\"932\">\n<p data-start=\"873\" data-end=\"932\">Use [latex]1 - 2\\sin^{2}\\theta[\/latex] if sine is easier.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"934\" data-end=\"1265\">\n<p data-start=\"937\" data-end=\"978\"><strong data-start=\"937\" data-end=\"976\">Work an Example with Special Angles<\/strong><\/p>\n<ul data-start=\"982\" data-end=\"1265\">\n<li data-start=\"982\" data-end=\"1040\">\n<p data-start=\"984\" data-end=\"1040\">Find [latex]\\sin(30^\\circ)[\/latex] using double-angle:<\/p>\n<\/li>\n<li data-start=\"1044\" data-end=\"1155\">\n<p data-start=\"1046\" data-end=\"1155\">[latex]\\sin(30^\\circ) = \\sin(2 \\cdot 15^\\circ)[\/latex]<br data-start=\"1100\" data-end=\"1103\" \/>= [latex]2\\sin(15^\\circ)\\cos(15^\\circ)[\/latex].<\/p>\n<\/li>\n<li data-start=\"1159\" data-end=\"1265\">\n<p data-start=\"1161\" data-end=\"1265\">Use known exact values of [latex]\\sin(15^\\circ)[\/latex] and [latex]\\cos(15^\\circ)[\/latex] to evaluate.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1267\" data-end=\"1417\">\n<p data-start=\"1270\" data-end=\"1297\"><strong data-start=\"1270\" data-end=\"1295\">Simplify Step by Step<\/strong><\/p>\n<ul data-start=\"1301\" data-end=\"1417\">\n<li data-start=\"1301\" data-end=\"1348\">\n<p data-start=\"1303\" data-end=\"1348\">Always reduce radicals and fractions fully.<\/p>\n<\/li>\n<li data-start=\"1352\" data-end=\"1417\">\n<p data-start=\"1354\" data-end=\"1417\">Keep answers exact in the form of fractions and square roots.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1419\" data-end=\"1613\">\n<p data-start=\"1422\" data-end=\"1446\"><strong data-start=\"1422\" data-end=\"1444\">Check the Quadrant<\/strong><\/p>\n<ul data-start=\"1450\" data-end=\"1613\">\n<li data-start=\"1450\" data-end=\"1613\">\n<p data-start=\"1452\" data-end=\"1613\">The sign of [latex]\\sin(2\\theta)[\/latex], [latex]\\cos(2\\theta)[\/latex], or [latex]\\tan(2\\theta)[\/latex] depends on which quadrant [latex]2\\theta[\/latex] is in.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1615\" data-end=\"1896\">\n<p data-start=\"1618\" data-end=\"1682\"><strong data-start=\"1618\" data-end=\"1629\">Example: <\/strong>Find [latex]\\cos(2 \\cdot 22.5^\\circ)[\/latex].<\/p>\n<ul data-start=\"1686\" data-end=\"1896\">\n<li data-start=\"1686\" data-end=\"1766\">\n<p data-start=\"1688\" data-end=\"1766\">[latex]\\cos(45^\\circ) = \\cos^{2}(22.5^\\circ) - \\sin^{2}(22.5^\\circ)[\/latex].<\/p>\n<\/li>\n<li data-start=\"1770\" data-end=\"1896\">\n<p data-start=\"1772\" data-end=\"1896\">Since [latex]\\cos(45^\\circ) = \\dfrac{\\sqrt{2}}{2}[\/latex], this connects the exact values of half-angles to double-angles.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">If [latex]\\sin\\theta = \\frac{3}{5}[\/latex] and [latex]\\theta[\/latex] is in Quadrant II, find [latex]\\sin(2\\theta)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qdouble-001\">Show Solution<\/button><\/p>\n<div id=\"qdouble-001\" class=\"hidden-answer\" style=\"display: none\">\nFirst, find [latex]\\cos\\theta[\/latex] using the Pythagorean identity:<br \/>\n[latex]\\begin{align*}  \\cos^2\\theta &= 1 - \\sin^2\\theta \\\\  &= 1 - \\left(\\frac{3}{5}\\right)^2 \\\\  &= 1 - \\frac{9}{25} \\\\  &= \\frac{16}{25}  \\end{align*}[\/latex]Since [latex]\\theta[\/latex] is in Quadrant II, [latex]\\cos\\theta = -\\frac{4}{5}[\/latex]Use the double-angle formula:<br \/>\n[latex]\\begin{align*}  \\sin(2\\theta) &= 2\\sin\\theta\\cos\\theta \\\\  &= 2\\left(\\frac{3}{5}\\right)\\left(-\\frac{4}{5}\\right) \\\\  &= -\\frac{24}{25}  \\end{align*}[\/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-cfdbdbcc-SE5SBTgrwH8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/SE5SBTgrwH8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfdbdbcc-SE5SBTgrwH8\" 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=14661367&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cfdbdbcc-SE5SBTgrwH8&#38;vembed=0&#38;video_id=SE5SBTgrwH8&#38;video_target=tpm-plugin-cfdbdbcc-SE5SBTgrwH8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Double+Angle+Identities+%26+Formulas+of+Sin%2C+Cos+%26+Tan+-+Trigonometry_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDouble Angle Identities &amp; Formulas of Sin, Cos &amp; Tan &#8211; Trigonometry\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Verifying Identities with Double-Angle Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"76\" data-end=\"469\">Double-angle formulas connect trigonometric functions of [latex]2\\theta[\/latex] to those of [latex]\\theta[\/latex]. They are powerful tools for proving that two trig expressions are equal. In verification problems, you expand one side using a double-angle identity, then simplify until it matches the other side. This shows that the identity holds for all values where both sides are defined.<\/p>\n<p data-start=\"471\" data-end=\"490\">The formulas are:<\/p>\n<ul data-start=\"492\" data-end=\"734\">\n<li data-start=\"492\" data-end=\"548\">\n<p data-start=\"494\" data-end=\"548\">[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"549\" data-end=\"659\">\n<p data-start=\"551\" data-end=\"659\">[latex]\\cos(2\\theta) = \\cos^{2}\\theta - \\sin^{2}\\theta = 2\\cos^{2}\\theta - 1 = 1 - 2\\sin^{2}\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"660\" data-end=\"734\">\n<p data-start=\"662\" data-end=\"734\">[latex]\\tan(2\\theta) = \\dfrac{2\\tan\\theta}{1 - \\tan^{2}\\theta}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Double-Angle Formulas in Verifications<\/strong><\/p>\n<ol>\n<li data-start=\"805\" data-end=\"1123\">\n<p data-start=\"808\" data-end=\"839\"><strong data-start=\"808\" data-end=\"837\">Choose the Right Identity<\/strong><\/p>\n<ul data-start=\"843\" data-end=\"1123\">\n<li data-start=\"843\" data-end=\"944\">\n<p data-start=\"845\" data-end=\"944\">If the expression involves [latex]\\sin\\theta\\cos\\theta[\/latex], use [latex]\\sin(2\\theta)[\/latex].<\/p>\n<\/li>\n<li data-start=\"948\" data-end=\"1062\">\n<p data-start=\"950\" data-end=\"1062\">If it involves [latex]\\cos^{2}\\theta[\/latex] or [latex]\\sin^{2}\\theta[\/latex], use one of the cosine formulas.<\/p>\n<\/li>\n<li data-start=\"1066\" data-end=\"1123\">\n<p data-start=\"1068\" data-end=\"1123\">If tangent appears, use [latex]\\tan(2\\theta)[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1125\" data-end=\"1274\">\n<p data-start=\"1128\" data-end=\"1164\"><strong data-start=\"1128\" data-end=\"1162\">Work From the Complicated Side<\/strong><\/p>\n<ul data-start=\"1168\" data-end=\"1274\">\n<li data-start=\"1168\" data-end=\"1227\">\n<p data-start=\"1170\" data-end=\"1227\">Expand or replace terms on the more complex side first.<\/p>\n<\/li>\n<li data-start=\"1231\" data-end=\"1274\">\n<p data-start=\"1233\" data-end=\"1274\">Leave the simpler side as the \u201ctarget.\u201d<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1276\" data-end=\"1481\">\n<p data-start=\"1279\" data-end=\"1306\"><strong data-start=\"1279\" data-end=\"1304\">Simplify Step by Step<\/strong><\/p>\n<ul data-start=\"1310\" data-end=\"1481\">\n<li data-start=\"1310\" data-end=\"1427\">\n<p data-start=\"1312\" data-end=\"1427\">Apply Pythagorean identities to replace [latex]\\sin^{2}\\theta[\/latex] or [latex]\\cos^{2}\\theta[\/latex] if needed.<\/p>\n<\/li>\n<li data-start=\"1431\" data-end=\"1481\">\n<p data-start=\"1433\" data-end=\"1481\">Combine fractions or cancel factors carefully.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Verify the identity: [latex]\\frac{\\sin(2\\theta)}{1 + \\cos(2\\theta)} = \\tan\\theta[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qdouble-002\">Show Solution<\/button><\/p>\n<div id=\"qdouble-002\" class=\"hidden-answer\" style=\"display: none\">\nWork with the left side:<br \/>\n[latex]\\begin{align*}  \\frac{\\sin(2\\theta)}{1 + \\cos(2\\theta)} &= \\frac{2\\sin\\theta\\cos\\theta}{1 + (2\\cos^2\\theta - 1)} && \\text{use double-angle formulas} \\\\  &= \\frac{2\\sin\\theta\\cos\\theta}{2\\cos^2\\theta} && \\text{simplify denominator} \\\\  &= \\frac{\\sin\\theta}{\\cos\\theta} && \\text{cancel } 2\\cos\\theta \\\\  &= \\tan\\theta && \\text{quotient identity}  \\end{align*}[\/latex]The identity is verified.\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-cacbbbfg-sZ1GjTqgR0s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sZ1GjTqgR0s?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cacbbbfg-sZ1GjTqgR0s\" 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=14661368&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cacbbbfg-sZ1GjTqgR0s&#38;vembed=0&#38;video_id=sZ1GjTqgR0s&#38;video_target=tpm-plugin-cacbbbfg-sZ1GjTqgR0s\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+verify+an+identity+using+double+angle+formulas_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to verify an identity using double angle formulas\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Simplifying Expressions with Reduction Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>Reduction formulas (also called power-reducing formulas) allow us to rewrite even powers of sine or cosine in terms of the first power of cosine. These formulas are derived from the double-angle formulas and are especially important in calculus.<\/p>\n<p>The reduction formulas are:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\sin^2\\theta = \\frac{1 - \\cos(2\\theta)}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cos^2\\theta = \\frac{1 + \\cos(2\\theta)}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan^2\\theta = \\frac{1 - \\cos(2\\theta)}{1 + \\cos(2\\theta)}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Using Reduction Formulas<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Apply the formula to reduce the power<\/li>\n<li class=\"whitespace-normal break-words\">If higher powers remain, apply the formula again<\/li>\n<li class=\"whitespace-normal break-words\">Simplify using algebra<\/li>\n<\/ul>\n<\/div>\n<div>\n<section class=\"textbox example\">Write an equivalent expression for [latex]\\sin^4 x[\/latex] that does not involve any powers of sine or cosine greater than 1. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qreduction-001\">Show Solution<\/button> <\/p>\n<div id=\"qreduction-001\" class=\"hidden-answer\" style=\"display: none\"> Apply the reduction formula twice: [latex]\\begin{align*} \\sin^4 x &= (\\sin^2 x)^2 \\\\ &= \\left(\\frac{1 - \\cos(2x)}{2}\\right)^2 && \\text{substitute reduction formula} \\\\ &= \\frac{1}{4}(1 - 2\\cos(2x) + \\cos^2(2x)) && \\text{expand} \\\\ &= \\frac{1}{4} - \\frac{1}{2}\\cos(2x) + \\frac{1}{4}\\left(\\frac{1 + \\cos(4x)}{2}\\right) && \\text{apply formula again} \\\\ &= \\frac{1}{4} - \\frac{1}{2}\\cos(2x) + \\frac{1}{8} + \\frac{1}{8}\\cos(4x) \\\\ &= \\frac{3}{8} - \\frac{1}{2}\\cos(2x) + \\frac{1}{8}\\cos(4x) \\end{align*}[\/latex] <\/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-dhageega-5Ipor4q0Jd8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5Ipor4q0Jd8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dhageega-5Ipor4q0Jd8\" 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=14661369&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dhageega-5Ipor4q0Jd8&#38;vembed=0&#38;video_id=5Ipor4q0Jd8&#38;video_target=tpm-plugin-dhageega-5Ipor4q0Jd8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Power+Reducing+Formulas+for+Sine+and+Cosine%2C+Example+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPower Reducing Formulas for Sine and Cosine, Example 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Half-Angle Formulas for Exact Values<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"65\" data-end=\"420\">Half-angle formulas allow us to evaluate trig functions of angles like [latex]\\dfrac{\\theta}{2}[\/latex] by rewriting them in terms of [latex]\\sin\\theta[\/latex] and [latex]\\cos\\theta[\/latex]. They are especially useful for finding exact values of angles not directly on the unit circle, such as [latex]22.5^\\circ[\/latex] or [latex]\\dfrac{\\pi}{8}[\/latex].<\/p>\n<p data-start=\"422\" data-end=\"441\">The formulas are:<\/p>\n<ul data-start=\"443\" data-end=\"746\">\n<li data-start=\"443\" data-end=\"532\">\n<p data-start=\"445\" data-end=\"532\">[latex]\\sin\\left(\\dfrac{\\theta}{2}\\right) = \\pm\\sqrt{\\dfrac{1-\\cos\\theta}{2}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"533\" data-end=\"622\">\n<p data-start=\"535\" data-end=\"622\">[latex]\\cos\\left(\\dfrac{\\theta}{2}\\right) = \\pm\\sqrt{\\dfrac{1+\\cos\\theta}{2}}[\/latex]<\/p>\n<\/li>\n<li data-start=\"623\" data-end=\"746\">\n<p data-start=\"625\" data-end=\"746\">[latex]\\tan\\left(\\dfrac{\\theta}{2}\\right) = \\dfrac{\\sin\\theta}{1+\\cos\\theta} = \\dfrac{1-\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"748\" data-end=\"842\">The sign (positive or negative) depends on the quadrant of [latex]\\dfrac{\\theta}{2}[\/latex].<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Using Half-Angle Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"894\" data-end=\"1069\">\n<p data-start=\"897\" data-end=\"921\"><strong data-start=\"897\" data-end=\"919\">Check the Quadrant<\/strong><\/p>\n<ul data-start=\"925\" data-end=\"1069\">\n<li data-start=\"925\" data-end=\"1010\">\n<p data-start=\"927\" data-end=\"1010\">Decide whether [latex]\\dfrac{\\theta}{2}[\/latex] is in Quadrant I, II, III, or IV.<\/p>\n<\/li>\n<li data-start=\"1014\" data-end=\"1069\">\n<p data-start=\"1016\" data-end=\"1069\">Use this to assign the correct sign in the formula.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1071\" data-end=\"1235\">\n<p data-start=\"1074\" data-end=\"1113\"><strong data-start=\"1074\" data-end=\"1111\">Choose the Formula That Fits Best<\/strong><\/p>\n<ul data-start=\"1117\" data-end=\"1235\">\n<li data-start=\"1117\" data-end=\"1168\">\n<p data-start=\"1119\" data-end=\"1168\">For sine and cosine, use the square root forms.<\/p>\n<\/li>\n<li data-start=\"1172\" data-end=\"1235\">\n<p data-start=\"1174\" data-end=\"1235\">For tangent, pick the version that avoids dividing by zero.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">Find the exact value of [latex]\\cos(15\u00b0)[\/latex] using a half-angle formula.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qhalf-001\">Show Solution<\/button><\/p>\n<div id=\"qhalf-001\" class=\"hidden-answer\" style=\"display: none\">\nWrite [latex]15\u00b0 = \\frac{30\u00b0}{2}[\/latex]Since [latex]15\u00b0[\/latex] is in Quadrant I, cosine is positive.Use the half-angle formula:<br \/>\n[latex]\\begin{align*}  \\cos(15\u00b0) &= \\cos\\left(\\frac{30\u00b0}{2}\\right) \\\\  &= +\\sqrt{\\frac{1 + \\cos(30\u00b0)}{2}} \\\\  &= \\sqrt{\\frac{1 + \\frac{\\sqrt{3}}{2}}{2}} \\\\  &= \\sqrt{\\frac{\\frac{2 + \\sqrt{3}}{2}}{2}} \\\\  &= \\sqrt{\\frac{2 + \\sqrt{3}}{4}} \\\\  &= \\frac{\\sqrt{2 + \\sqrt{3}}}{2}  \\end{align*}[\/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-fadhehdg-ZncEctA2Fug\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ZncEctA2Fug?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fadhehdg-ZncEctA2Fug\" 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=14661370&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fadhehdg-ZncEctA2Fug&#38;vembed=0&#38;video_id=ZncEctA2Fug&#38;video_target=tpm-plugin-fadhehdg-ZncEctA2Fug\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Evaluate+the+half+angle+in+radians+for+sine_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate the half angle in radians for sine\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Double Angle Identities & Formulas of Sin, Cos & Tan - 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