{"id":259362,"date":"2025-07-19T00:19:51","date_gmt":"2025-07-19T00:19:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259362"},"modified":"2025-07-19T00:19:53","modified_gmt":"2025-07-19T00:19:53","slug":"find-the-exact-value-ofcos%e2%81%a1%e2%88%92195%e2%88%98-by-using-a-sum-or-difference-formula","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/find-the-exact-value-ofcos%e2%81%a1%e2%88%92195%e2%88%98-by-using-a-sum-or-difference-formula\/","title":{"rendered":"Find the exact value ofcos\u2061(\u2212195\u2218) \u00a0\u00a0by using a sum or difference formula."},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-542.png\" alt=\"\" class=\"wp-image-259363\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the exact value of cos\u2061(\u2212195\u2218)\\cos(-195^\\circ)cos(\u2212195\u2218) using a sum or difference formula, we can break the angle into two components whose cosine values are easier to calculate. Let&#8217;s use the sum formula for cosine: cos\u2061(A+B)=cos\u2061(A)cos\u2061(B)\u2212sin\u2061(A)sin\u2061(B)\\cos(A + B) = \\cos(A)\\cos(B) &#8211; \\sin(A)\\sin(B)cos(A+B)=cos(A)cos(B)\u2212sin(A)sin(B)<\/p>\n\n\n\n<p>In this case, we can express \u2212195\u2218-195^\\circ\u2212195\u2218 as a sum of \u2212180\u2218-180^\\circ\u2212180\u2218 and \u221215\u2218-15^\\circ\u221215\u2218 because these angles are easier to work with: \u2212195\u2218=\u2212180\u2218\u221215\u2218-195^\\circ = -180^\\circ &#8211; 15^\\circ\u2212195\u2218=\u2212180\u2218\u221215\u2218<\/p>\n\n\n\n<p>Now, apply the cosine sum formula: cos\u2061(\u2212195\u2218)=cos\u2061(\u2212180\u2218\u221215\u2218)=cos\u2061(\u2212180\u2218)cos\u2061(\u221215\u2218)\u2212sin\u2061(\u2212180\u2218)sin\u2061(\u221215\u2218)\\cos(-195^\\circ) = \\cos(-180^\\circ &#8211; 15^\\circ) = \\cos(-180^\\circ)\\cos(-15^\\circ) &#8211; \\sin(-180^\\circ)\\sin(-15^\\circ)cos(\u2212195\u2218)=cos(\u2212180\u2218\u221215\u2218)=cos(\u2212180\u2218)cos(\u221215\u2218)\u2212sin(\u2212180\u2218)sin(\u221215\u2218)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Find the trigonometric values for \u2212180\u2218-180^\\circ\u2212180\u2218 and \u221215\u2218-15^\\circ\u221215\u2218<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cos\u2061(\u2212180\u2218)=\u22121\\cos(-180^\\circ) = -1cos(\u2212180\u2218)=\u22121 (since cosine is an even function)<\/li>\n\n\n\n<li>sin\u2061(\u2212180\u2218)=0\\sin(-180^\\circ) = 0sin(\u2212180\u2218)=0 (since sine is an odd function)<\/li>\n<\/ul>\n\n\n\n<p>Now for \u221215\u2218-15^\\circ\u221215\u2218, we know the standard values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cos\u2061(\u221215\u2218)=cos\u2061(15\u2218)\\cos(-15^\\circ) = \\cos(15^\\circ)cos(\u221215\u2218)=cos(15\u2218) (since cosine is an even function)<\/li>\n\n\n\n<li>sin\u2061(\u221215\u2218)=\u2212sin\u2061(15\u2218)\\sin(-15^\\circ) = -\\sin(15^\\circ)sin(\u221215\u2218)=\u2212sin(15\u2218) (since sine is an odd function)<\/li>\n<\/ul>\n\n\n\n<p>Thus, we need the exact values for cos\u2061(15\u2218)\\cos(15^\\circ)cos(15\u2218) and sin\u2061(15\u2218)\\sin(15^\\circ)sin(15\u2218), which are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cos\u2061(15\u2218)=6+24\\cos(15^\\circ) = \\frac{\\sqrt{6} + \\sqrt{2}}{4}cos(15\u2218)=46\u200b+2\u200b\u200b<\/li>\n\n\n\n<li>sin\u2061(15\u2218)=6\u221224\\sin(15^\\circ) = \\frac{\\sqrt{6} &#8211; \\sqrt{2}}{4}sin(15\u2218)=46\u200b\u22122\u200b\u200b<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Substitute into the sum formula<\/h3>\n\n\n\n<p>cos\u2061(\u2212195\u2218)=(\u22121)\u22c5(6+24)\u22120\u22c5(6\u221224)\\cos(-195^\\circ) = (-1) \\cdot \\left( \\frac{\\sqrt{6} + \\sqrt{2}}{4} \\right) &#8211; 0 \\cdot \\left( \\frac{\\sqrt{6} &#8211; \\sqrt{2}}{4} \\right)cos(\u2212195\u2218)=(\u22121)\u22c5(46\u200b+2\u200b\u200b)\u22120\u22c5(46\u200b\u22122\u200b\u200b)<\/p>\n\n\n\n<p>Simplifying: cos\u2061(\u2212195\u2218)=\u22126+24\\cos(-195^\\circ) = &#8211; \\frac{\\sqrt{6} + \\sqrt{2}}{4}cos(\u2212195\u2218)=\u221246\u200b+2\u200b\u200b<\/p>\n\n\n\n<p>Thus, the exact value of cos\u2061(\u2212195\u2218)\\cos(-195^\\circ)cos(\u2212195\u2218) is: \u22126+24\\boxed{-\\frac{\\sqrt{6} + \\sqrt{2}}{4}}\u221246\u200b+2\u200b\u200b\u200b<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-947.jpeg\" alt=\"\" class=\"wp-image-259364\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The Correct Answer and Explanation is: To find the exact value of cos\u2061(\u2212195\u2218)\\cos(-195^\\circ)cos(\u2212195\u2218) using a sum or difference formula, we can break the angle into two components whose cosine values are easier to calculate. Let&#8217;s use the sum formula for cosine: cos\u2061(A+B)=cos\u2061(A)cos\u2061(B)\u2212sin\u2061(A)sin\u2061(B)\\cos(A + B) = \\cos(A)\\cos(B) &#8211; \\sin(A)\\sin(B)cos(A+B)=cos(A)cos(B)\u2212sin(A)sin(B) In this case, we can express \u2212195\u2218-195^\\circ\u2212195\u2218 [&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-259362","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259362","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=259362"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259362\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259362"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259362"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259362"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}