{"id":184778,"date":"2025-01-21T15:30:54","date_gmt":"2025-01-21T15:30:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=184778"},"modified":"2025-01-21T15:30:55","modified_gmt":"2025-01-21T15:30:55","slug":"find-the-exact-value-of-cos-22-5-degree-using-the-half-angle-identity","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/21\/find-the-exact-value-of-cos-22-5-degree-using-the-half-angle-identity\/","title":{"rendered":"Find the exact value of cos 22.5 degree using the half-angle identity"},"content":{"rendered":"\n

Find the exact value of cos 22.5 degree using the half-angle identity.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To find the exact value of cos\u2061(22.5\u2218)\\cos(22.5^\\circ), we use the half-angle identity for cosine<\/strong>, which is: cos\u2061(\u03b82)=\u00b11+cos\u2061(\u03b8)2\\cos\\left(\\frac{\\theta}{2}\\right) = \\pm \\sqrt{\\frac{1 + \\cos(\\theta)}{2}}<\/p>\n\n\n\n

Step-by-step solution:<\/h3>\n\n\n\n
    \n
  1. Identify the angle and the quadrant<\/strong>:\n
      \n
    • Here, \u03b8=45\u2218\\theta = 45^\\circ, and we are finding cos\u2061(\u03b82)=cos\u2061(22.5\u2218)\\cos\\left(\\frac{\\theta}{2}\\right) = \\cos(22.5^\\circ).<\/li>\n\n\n\n
    • Since 22.5\u221822.5^\\circ is in the first quadrant, cos\u2061(22.5\u2218)>0\\cos(22.5^\\circ) > 0, so we take the positive square root.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Use the half-angle identity<\/strong>: cos\u2061(22.5\u2218)=1+cos\u2061(45\u2218)2\\cos(22.5^\\circ) = \\sqrt{\\frac{1 + \\cos(45^\\circ)}{2}}<\/li>\n\n\n\n
    • Substitute cos\u2061(45\u2218)\\cos(45^\\circ)<\/strong>: From trigonometric values, cos\u2061(45\u2218)=22\\cos(45^\\circ) = \\frac{\\sqrt{2}}{2}. Substituting this into the formula: cos\u2061(22.5\u2218)=1+222\\cos(22.5^\\circ) = \\sqrt{\\frac{1 + \\frac{\\sqrt{2}}{2}}{2}}<\/li>\n\n\n\n
    • Simplify the expression<\/strong>: Combine terms inside the numerator: cos\u2061(22.5\u2218)=22+222\\cos(22.5^\\circ) = \\sqrt{\\frac{\\frac{2}{2} + \\frac{\\sqrt{2}}{2}}{2}} cos\u2061(22.5\u2218)=2+222\\cos(22.5^\\circ) = \\sqrt{\\frac{\\frac{2 + \\sqrt{2}}{2}}{2}} cos\u2061(22.5\u2218)=2+24\\cos(22.5^\\circ) = \\sqrt{\\frac{2 + \\sqrt{2}}{4}}<\/li>\n\n\n\n
    • Final simplification<\/strong>: cos\u2061(22.5\u2218)=2+22\\cos(22.5^\\circ) = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}<\/li>\n<\/ol>\n\n\n\n

      Explanation:<\/h3>\n\n\n\n

      The key to solving this is the half-angle identity<\/strong> and knowing the exact value of cos\u2061(45\u2218)\\cos(45^\\circ). By substituting and carefully simplifying the fractions, we find that: cos\u2061(22.5\u2218)=2+22\\cos(22.5^\\circ) = \\frac{\\sqrt{2 + \\sqrt{2}}}{2}<\/p>\n\n\n\n

      This exact form highlights the importance of simplifying nested fractions and square roots systematically.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Find the exact value of cos 22.5 degree using the half-angle identity. The Correct Answer and Explanation is : To find the exact value of cos\u2061(22.5\u2218)\\cos(22.5^\\circ), we use the half-angle identity for cosine, which is: cos\u2061(\u03b82)=\u00b11+cos\u2061(\u03b8)2\\cos\\left(\\frac{\\theta}{2}\\right) = \\pm \\sqrt{\\frac{1 + \\cos(\\theta)}{2}} Step-by-step solution: Explanation: The key to solving this is the half-angle identity and knowing […]<\/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-184778","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184778","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=184778"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184778\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184778"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}