{"id":234241,"date":"2025-06-14T02:47:59","date_gmt":"2025-06-14T02:47:59","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234241"},"modified":"2025-06-14T02:48:01","modified_gmt":"2025-06-14T02:48:01","slug":"find-the-exact-value-of-cos-75-degrees-using-sum-or-difference-angle-identities","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/find-the-exact-value-of-cos-75-degrees-using-sum-or-difference-angle-identities\/","title":{"rendered":"find the exact value of cos 75 degrees using sum or difference angle identities"},"content":{"rendered":"\n<p>find the exact value of cos 75 degrees using sum or difference angle identities<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the exact value of <strong>cos(75\u00b0)<\/strong> using sum or difference identities, we express 75\u00b0 as the <strong>sum of two known angles<\/strong> with known cosine and sine values.<br>We know: 75\u2218=45\u2218+30\u221875^\\circ = 45^\\circ + 30^\\circ75\u2218=45\u2218+30\u2218<\/p>\n\n\n\n<p>Now apply the <strong>cosine of sum identity<\/strong>: cos\u2061(A+B)=cos\u2061Acos\u2061B\u2212sin\u2061Asin\u2061B\\cos(A + B) = \\cos A \\cos B &#8211; \\sin A \\sin Bcos(A+B)=cosAcosB\u2212sinAsinB<\/p>\n\n\n\n<p>Substitute A=45\u2218A = 45^\\circA=45\u2218 and B=30\u2218B = 30^\\circB=30\u2218: cos\u2061(75\u2218)=cos\u2061(45\u2218+30\u2218)=cos\u206145\u2218cos\u206130\u2218\u2212sin\u206145\u2218sin\u206130\u2218\\cos(75^\\circ) = \\cos(45^\\circ + 30^\\circ) = \\cos 45^\\circ \\cos 30^\\circ &#8211; \\sin 45^\\circ \\sin 30^\\circcos(75\u2218)=cos(45\u2218+30\u2218)=cos45\u2218cos30\u2218\u2212sin45\u2218sin30\u2218<\/p>\n\n\n\n<p>Recall the exact trigonometric values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cos\u206145\u2218=22\\cos 45^\\circ = \\frac{\\sqrt{2}}{2}cos45\u2218=22\u200b\u200b<\/li>\n\n\n\n<li>sin\u206145\u2218=22\\sin 45^\\circ = \\frac{\\sqrt{2}}{2}sin45\u2218=22\u200b\u200b<\/li>\n\n\n\n<li>cos\u206130\u2218=32\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}cos30\u2218=23\u200b\u200b<\/li>\n\n\n\n<li>sin\u206130\u2218=12\\sin 30^\\circ = \\frac{1}{2}sin30\u2218=21\u200b<\/li>\n<\/ul>\n\n\n\n<p>Substitute these into the identity: cos\u2061(75\u2218)=(22)(32)\u2212(22)(12)\\cos(75^\\circ) = \\left(\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{\\sqrt{3}}{2}\\right) &#8211; \\left(\\frac{\\sqrt{2}}{2}\\right)\\left(\\frac{1}{2}\\right)cos(75\u2218)=(22\u200b\u200b)(23\u200b\u200b)\u2212(22\u200b\u200b)(21\u200b) cos\u2061(75\u2218)=64\u221224\\cos(75^\\circ) = \\frac{\\sqrt{6}}{4} &#8211; \\frac{\\sqrt{2}}{4}cos(75\u2218)=46\u200b\u200b\u221242\u200b\u200b cos\u2061(75\u2218)=6\u221224\\cos(75^\\circ) = \\frac{\\sqrt{6} &#8211; \\sqrt{2}}{4}cos(75\u2218)=46\u200b\u22122\u200b\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>cos\u2061(75\u2218)=6\u221224\\boxed{\\cos(75^\\circ) = \\frac{\\sqrt{6} &#8211; \\sqrt{2}}{4}}cos(75\u2218)=46\u200b\u22122\u200b\u200b\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This problem utilizes a fundamental trigonometric identity to evaluate a non-standard angle. Since 75\u00b0 is not a common angle on the unit circle, it must be expressed as a sum or difference of angles with known trigonometric values. The cosine of the sum identity provides a reliable method for calculating the exact value. After identifying 75\u00b0 as 45\u00b0 + 30\u00b0, the identity is applied by substituting in the known values for cosine and sine at those angles. The result simplifies to a rational expression involving square roots, providing the exact trigonometric value of cosine at 75 degrees without needing a calculator.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-503.jpeg\" alt=\"\" class=\"wp-image-234242\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>find the exact value of cos 75 degrees using sum or difference angle identities The Correct Answer and Explanation is: To find the exact value of cos(75\u00b0) using sum or difference identities, we express 75\u00b0 as the sum of two known angles with known cosine and sine values.We know: 75\u2218=45\u2218+30\u221875^\\circ = 45^\\circ + 30^\\circ75\u2218=45\u2218+30\u2218 Now [&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-234241","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234241","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=234241"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234241\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234241"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234241"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}