{"id":204446,"date":"2025-03-21T20:59:47","date_gmt":"2025-03-21T20:59:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=204446"},"modified":"2025-03-21T20:59:49","modified_gmt":"2025-03-21T20:59:49","slug":"find-the-expression-as-the-sine-or-cosine-of-an-angle-cos-100-degree-cos-50-degree-sin-100-degrees-in-50-degree","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/21\/find-the-expression-as-the-sine-or-cosine-of-an-angle-cos-100-degree-cos-50-degree-sin-100-degrees-in-50-degree\/","title":{"rendered":"Find the expression as the sine or cosine of an angle, cos 100 Degree cos 50 Degree &#8211; sin 100 Degrees in 50 Degree"},"content":{"rendered":"\n<p>Find the expression as the sine or cosine of an angle, cos 100 Degree cos 50 Degree &#8211; sin 100 Degrees in 50 Degree<\/p>\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>The given expression is:<\/p>\n\n\n\n<p>[<br>\\cos 100^\\circ \\cos 50^\\circ &#8211; \\sin 100^\\circ \\sin 50^\\circ<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Recognizing the Identity<\/h3>\n\n\n\n<p>We recognize that this expression matches the <strong>cosine addition formula<\/strong>:<\/p>\n\n\n\n<p>[<br>\\cos(A + B) = \\cos A \\cos B &#8211; \\sin A \\sin B<br>]<\/p>\n\n\n\n<p>Comparing the given expression with the formula, we see that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( A = 100^\\circ )<\/li>\n\n\n\n<li>( B = 50^\\circ )<\/li>\n<\/ul>\n\n\n\n<p>So, we can rewrite the expression as:<\/p>\n\n\n\n<p>[<br>\\cos(100^\\circ + 50^\\circ)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Simplifying<\/h3>\n\n\n\n<p>[<br>\\cos(150^\\circ)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Finding the Exact Value<\/h3>\n\n\n\n<p>From trigonometric values, we know:<\/p>\n\n\n\n<p>[<br>\\cos 150^\\circ = -\\cos 30^\\circ<br>]<\/p>\n\n\n\n<p>Since ( \\cos 30^\\circ = \\frac{\\sqrt{3}}{2} ), we get:<\/p>\n\n\n\n<p>[<br>\\cos 150^\\circ = -\\frac{\\sqrt{3}}{2}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>Thus, the original expression simplifies to:<\/p>\n\n\n\n<p>[<br>-\\frac{\\sqrt{3}}{2}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300 Words)<\/h3>\n\n\n\n<p>The given trigonometric expression involves cosine and sine functions, and it closely follows the pattern of the cosine sum identity:<\/p>\n\n\n\n<p>[<br>\\cos(A + B) = \\cos A \\cos B &#8211; \\sin A \\sin B<br>]<\/p>\n\n\n\n<p>By recognizing this identity, we substitute ( A = 100^\\circ ) and ( B = 50^\\circ ), leading us to:<\/p>\n\n\n\n<p>[<br>\\cos(100^\\circ + 50^\\circ) = \\cos 150^\\circ<br>]<\/p>\n\n\n\n<p>Since ( \\cos 150^\\circ ) is known from standard trigonometric values, we express it in terms of a known angle:<\/p>\n\n\n\n<p>[<br>\\cos 150^\\circ = -\\cos 30^\\circ<br>]<\/p>\n\n\n\n<p>With ( \\cos 30^\\circ = \\frac{\\sqrt{3}}{2} ), this simplifies further to:<\/p>\n\n\n\n<p>[<br>-\\frac{\\sqrt{3}}{2}<br>]<\/p>\n\n\n\n<p>This method is an application of fundamental trigonometric identities, helping to simplify complex expressions efficiently.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the expression as the sine or cosine of an angle, cos 100 Degree cos 50 Degree &#8211; sin 100 Degrees in 50 Degree The correct answer and explanation is : The given expression is: [\\cos 100^\\circ \\cos 50^\\circ &#8211; \\sin 100^\\circ \\sin 50^\\circ] Step 1: Recognizing the Identity We recognize that this expression matches [&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 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