{"id":182588,"date":"2025-01-14T12:22:01","date_gmt":"2025-01-14T12:22:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182588"},"modified":"2025-01-14T12:22:03","modified_gmt":"2025-01-14T12:22:03","slug":"find-the-exact-value-of-cos-3-pi-8-cos-pi-8","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/14\/find-the-exact-value-of-cos-3-pi-8-cos-pi-8\/","title":{"rendered":"Find The Exact Value Of Cos 3 Pi\/8 + Cos Pi\/8\u00a0"},"content":{"rendered":"\n<p>Find The Exact Value Of Cos 3 Pi\/8 + Cos Pi\/8\u00a0<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-192.png\" alt=\"\" class=\"wp-image-182589\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To find the exact value of ( \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) ), we can utilize trigonometric identities and known values of cosine at specific angles.<\/p>\n\n\n\n<p><strong>Step 1: Recognize Known Values<\/strong><\/p>\n\n\n\n<p>We start by recalling the exact values of cosine for certain standard angles:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\cos\\left(\\frac{\\pi}{8}\\right) = \\sqrt{\\frac{2 + \\sqrt{2}}{4}} )<\/li>\n\n\n\n<li>( \\cos\\left(\\frac{3\\pi}{8}\\right) = \\sqrt{\\frac{2 &#8211; \\sqrt{2}}{4}} )<\/li>\n<\/ul>\n\n\n\n<p>These values are derived from half-angle identities and are well-established in trigonometric tables.<\/p>\n\n\n\n<p><strong>Step 2: Apply the Sum of Cosines Identity<\/strong><\/p>\n\n\n\n<p>The sum of cosines identity states:<\/p>\n\n\n\n<p>[ \\cos A + \\cos B = 2 \\cos\\left(\\frac{A + B}{2}\\right) \\cdot \\cos\\left(\\frac{A &#8211; B}{2}\\right) ]<\/p>\n\n\n\n<p>Let ( A = \\frac{3\\pi}{8} ) and ( B = \\frac{\\pi}{8} ). Applying the identity:<\/p>\n\n\n\n<p>[ \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) = 2 \\cos\\left(\\frac{\\frac{3\\pi}{8} + \\frac{\\pi}{8}}{2}\\right) \\cdot \\cos\\left(\\frac{\\frac{3\\pi}{8} &#8211; \\frac{\\pi}{8}}{2}\\right) ]<\/p>\n\n\n\n<p>Simplifying the angles:<\/p>\n\n\n\n<p>[ = 2 \\cos\\left(\\frac{4\\pi}{16}\\right) \\cdot \\cos\\left(\\frac{2\\pi}{16}\\right) ]<\/p>\n\n\n\n<p>[ = 2 \\cos\\left(\\frac{\\pi}{4}\\right) \\cdot \\cos\\left(\\frac{\\pi}{8}\\right) ]<\/p>\n\n\n\n<p><strong>Step 3: Substitute Known Values<\/strong><\/p>\n\n\n\n<p>We know that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\cos\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2} )<\/li>\n\n\n\n<li>( \\cos\\left(\\frac{\\pi}{8}\\right) = \\sqrt{\\frac{2 + \\sqrt{2}}{4}} )<\/li>\n<\/ul>\n\n\n\n<p>Substituting these into the equation:<\/p>\n\n\n\n<p>[ \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) = 2 \\times \\frac{\\sqrt{2}}{2} \\times \\sqrt{\\frac{2 + \\sqrt{2}}{4}} ]<\/p>\n\n\n\n<p>Simplifying:<\/p>\n\n\n\n<p>[ = \\sqrt{2} \\times \\sqrt{\\frac{2 + \\sqrt{2}}{4}} ]<\/p>\n\n\n\n<p>[ = \\sqrt{\\frac{2 \\times (2 + \\sqrt{2})}{4}} ]<\/p>\n\n\n\n<p>[ = \\sqrt{\\frac{4 + 2\\sqrt{2}}{4}} ]<\/p>\n\n\n\n<p>[ = \\sqrt{1 + \\frac{\\sqrt{2}}{2}} ]<\/p>\n\n\n\n<p><strong>Step 4: Final Simplification<\/strong><\/p>\n\n\n\n<p>[ = \\sqrt{\\frac{2 + \\sqrt{2}}{2}} ]<\/p>\n\n\n\n<p>Therefore, the exact value of ( \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) ) is ( \\sqrt{\\frac{2 + \\sqrt{2}}{2}} ).<\/p>\n\n\n\n<p><strong>Conclusion<\/strong><\/p>\n\n\n\n<p>By applying trigonometric identities and utilizing known values of cosine at specific angles, we have determined that the exact value of ( \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) ) is ( \\sqrt{\\frac{2 + \\sqrt{2}}{2}} ).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find The Exact Value Of Cos 3 Pi\/8 + Cos Pi\/8\u00a0 The Correct Answer and Explanation is : To find the exact value of ( \\cos\\left(\\frac{3\\pi}{8}\\right) + \\cos\\left(\\frac{\\pi}{8}\\right) ), we can utilize trigonometric identities and known values of cosine at specific angles. Step 1: Recognize Known Values We start by recalling the exact values of [&hellip;]<\/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-182588","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182588","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=182588"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182588\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182588"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182588"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}