To find the exact value of cos(\u22123\u03c0\/8)<\/strong> using a half-angle identity, we follow these steps:<\/p>\n\n\n\n The half-angle identity for cosine is:cos\u2061(\u03b82)=\u00b11+cos\u2061(\u03b8)2\\cos\\left(\\frac{\\theta}{2}\\right) = \\pm \\sqrt{\\frac{1 + \\cos(\\theta)}{2}}cos(2\u03b8\u200b)=\u00b121+cos(\u03b8)\u200b\u200b<\/p>\n\n\n\n The sign depends on the quadrant in which the angle lies.<\/p>\n\n\n\n Let\u03b82=\u22123\u03c08\u21d2\u03b8=\u22123\u03c04\\frac{\\theta}{2} = -\\frac{3\\pi}{8} \\Rightarrow \\theta = -\\frac{3\\pi}{4}2\u03b8\u200b=\u221283\u03c0\u200b\u21d2\u03b8=\u221243\u03c0\u200b<\/p>\n\n\n\n So we will use:cos\u2061(\u22123\u03c08)=\u00b11+cos\u2061(\u22123\u03c04)2\\cos\\left(-\\frac{3\\pi}{8}\\right) = \\pm \\sqrt{\\frac{1 + \\cos\\left(-\\frac{3\\pi}{4}\\right)}{2}}cos(\u221283\u03c0\u200b)=\u00b121+cos(\u221243\u03c0\u200b)\u200b\u200b<\/p>\n\n\n\n Use the even property of cosine:cos\u2061(\u22123\u03c04)=cos\u2061(3\u03c04)\\cos\\left(-\\frac{3\\pi}{4}\\right) = \\cos\\left(\\frac{3\\pi}{4}\\right)cos(\u221243\u03c0\u200b)=cos(43\u03c0\u200b)cos\u2061(3\u03c04)=\u221222\\cos\\left(\\frac{3\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}cos(43\u03c0\u200b)=\u221222\u200b\u200b<\/p>\n\n\n\n cos\u2061(\u22123\u03c08)=1+(\u221222)2=1\u2212222\\cos\\left(-\\frac{3\\pi}{8}\\right) = \\sqrt{\\frac{1 + \\left(-\\frac{\\sqrt{2}}{2}\\right)}{2}} = \\sqrt{\\frac{1 – \\frac{\\sqrt{2}}{2}}{2}}cos(\u221283\u03c0\u200b)=21+(\u221222\u200b\u200b)\u200b\u200b=21\u221222\u200b\u200b\u200b\u200b=2\u221224=2\u221222= \\sqrt{\\frac{2 – \\sqrt{2}}{4}} = \\frac{\\sqrt{2 – \\sqrt{2}}}{2}=42\u22122\u200b\u200b\u200b=22\u22122\u200b\u200b\u200b<\/p>\n\n\n\n Now determine the correct sign. Since \u22123\u03c0\/8 is in the fourth quadrant, cosine is positive<\/strong> there.<\/p>\n\n\n\n cos\u2061(\u22123\u03c08)=2\u221222\\cos\\left(-\\frac{3\\pi}{8}\\right) = \\frac{\\sqrt{2 – \\sqrt{2}}}{2}cos(\u221283\u03c0\u200b)=22\u22122\u200b\u200b\u200b<\/p>\n\n\n\n To solve trigonometric expressions exactly, identities such as the half-angle formula are essential. The half-angle formula for cosine simplifies the cosine of an angle that is half of a known angle. In this case, \u22123\u03c0\/8 is half of \u22123\u03c0\/4. By applying the identity, we were able to rewrite the expression in terms of a more familiar angle: 3\u03c0\/4. The cosine of 3\u03c0\/4 is a standard value from the unit circle. Substituting this value into the formula leads to a square root expression. Because \u22123\u03c0\/8 lies in the fourth quadrant where cosine is positive, the positive root was selected. This method ensures that we maintain the exact trigonometric value rather than estimating or using a calculator.<\/p>\n\n\n\n Use half angle identity to find cos (-3pi\/8) The Correct Answer and Explanation is: To find the exact value of cos(\u22123\u03c0\/8) using a half-angle identity, we follow these steps: Step 1: Recognize the half-angle identity for cosine The half-angle identity for cosine is:cos\u2061(\u03b82)=\u00b11+cos\u2061(\u03b8)2\\cos\\left(\\frac{\\theta}{2}\\right) = \\pm \\sqrt{\\frac{1 + \\cos(\\theta)}{2}}cos(2\u03b8\u200b)=\u00b121+cos(\u03b8)\u200b\u200b The sign depends on the quadrant in […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-238174","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238174","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=238174"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238174\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238174"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238174"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Recognize the half-angle identity for cosine<\/h3>\n\n\n\n
\n\n\n\nStep 2: Express \u22123\u03c0\/8 as a half-angle<\/h3>\n\n\n\n
\n\n\n\nStep 3: Evaluate cosine of the double angle<\/h3>\n\n\n\n
\n\n\n\nStep 4: Plug in the value<\/h3>\n\n\n\n
\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation:<\/h3>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-996.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"