{"id":244245,"date":"2025-07-04T22:13:56","date_gmt":"2025-07-04T22:13:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244245"},"modified":"2025-07-04T22:13:58","modified_gmt":"2025-07-04T22:13:58","slug":"sin%ce%b8-and-cos%e2%81%a1%cf%953-8-both-angles-terminate-in-quadrant-iv-find-the-exact-value-of-sin%e2%81%a1%ce%b8%e2%88%92%cf%95","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/sin%ce%b8-and-cos%e2%81%a1%cf%953-8-both-angles-terminate-in-quadrant-iv-find-the-exact-value-of-sin%e2%81%a1%ce%b8%e2%88%92%cf%95\/","title":{"rendered":"\u00a0sin\u03b8 =and\u00a0cos\u2061\u03d5=3 \/8, both angles terminate in Quadrant IV find the exact value of\u00a0sin\u2061(\u03b8\u2212\u03d5).\u00a0"},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">\u00a0sin\u03b8  =and\u00a0cos\u2061\u03d5=3 \/8, both angles terminate in Quadrant IV find the exact value of\u00a0sin\u2061(\u03b8\u2212\u03d5).\u00a0<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-161.png\" alt=\"\" class=\"wp-image-244246\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the exact value of sin(\u03b8 \u2014 \u03c6), we will use the sine difference identity:<br>sin(\u03b8 \u2014 \u03c6) = sin(\u03b8)cos(\u03c6) \u2014 cos(\u03b8)sin(\u03c6)<\/p>\n\n\n\n<p>We are given the following information:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin(\u03b8) = -5\/7<\/li>\n\n\n\n<li>cos(\u03c6) = 3\/8<\/li>\n\n\n\n<li>Both angles \u03b8 and \u03c6 are in Quadrant IV.<\/li>\n<\/ul>\n\n\n\n<p>We need to find the values of cos(\u03b8) and sin(\u03c6).<\/p>\n\n\n\n<p><strong>Step 1: Find cos(\u03b8)<\/strong><br>We use the Pythagorean identity sin\u00b2(\u03b8) + cos\u00b2(\u03b8) = 1.<br>(-5\/7)\u00b2 + cos\u00b2(\u03b8) = 1<br>25\/49 + cos\u00b2(\u03b8) = 1<br>cos\u00b2(\u03b8) = 1 &#8211; 25\/49<br>cos\u00b2(\u03b8) = 49\/49 &#8211; 25\/49<br>cos\u00b2(\u03b8) = 24\/49<br>cos(\u03b8) = \u00b1\u221a(24\/49) = \u00b1\u221a24 \/ 7<\/p>\n\n\n\n<p>Since \u03b8 is in Quadrant IV, its cosine value is positive.<br>Therefore, cos(\u03b8) = \u221a24 \/ 7.<\/p>\n\n\n\n<p><strong>Step 2: Find sin(\u03c6)<\/strong><br>We use the Pythagorean identity again: sin\u00b2(\u03c6) + cos\u00b2(\u03c6) = 1.<br>sin\u00b2(\u03c6) + (3\/8)\u00b2 = 1<br>sin\u00b2(\u03c6) + 9\/64 = 1<br>sin\u00b2(\u03c6) = 1 &#8211; 9\/64<br>sin\u00b2(\u03c6) = 64\/64 &#8211; 9\/64<br>sin\u00b2(\u03c6) = 55\/64<br>sin(\u03c6) = \u00b1\u221a(55\/64) = \u00b1\u221a55 \/ 8<\/p>\n\n\n\n<p>Since \u03c6 is in Quadrant IV, its sine value is negative.<br>Therefore, sin(\u03c6) = -\u221a55 \/ 8.<\/p>\n\n\n\n<p><strong>Step 3: Substitute the values into the sine difference formula<\/strong><br>Now we have all the necessary components:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin(\u03b8) = -5\/7<\/li>\n\n\n\n<li>cos(\u03c6) = 3\/8<\/li>\n\n\n\n<li>cos(\u03b8) = \u221a24 \/ 7<\/li>\n\n\n\n<li>sin(\u03c6) = -\u221a55 \/ 8<\/li>\n<\/ul>\n\n\n\n<p>Substitute these into the formula sin(\u03b8 \u2014 \u03c6) = sin(\u03b8)cos(\u03c6) \u2014 cos(\u03b8)sin(\u03c6):<br>sin(\u03b8 \u2014 \u03c6) = (-5\/7)(3\/8) \u2014 (\u221a24 \/ 7)(-\u221a55 \/ 8)<br>sin(\u03b8 \u2014 \u03c6) = -15\/56 \u2014 (-\u221a(24 * 55) \/ 56)<br>sin(\u03b8 \u2014 \u03c6) = -15\/56 + \u221a1320 \/ 56<br>sin(\u03b8 \u2014 \u03c6) = (-15 + \u221a1320) \/ 56<\/p>\n\n\n\n<p>The problem states not to simplify radicals, so this is the final exact form.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>sin(\u03b8 \u2014 \u03c6)\u00a0=\u00a0<strong>(-15 + \u221a1320)\/56<\/strong><\/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-banner5-411.jpeg\" alt=\"\" class=\"wp-image-244247\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0sin\u03b8 =and\u00a0cos\u2061\u03d5=3 \/8, both angles terminate in Quadrant IV find the exact value of\u00a0sin\u2061(\u03b8\u2212\u03d5).\u00a0 The Correct Answer and Explanation is: To find the exact value of sin(\u03b8 \u2014 \u03c6), we will use the sine difference identity:sin(\u03b8 \u2014 \u03c6) = sin(\u03b8)cos(\u03c6) \u2014 cos(\u03b8)sin(\u03c6) We are given the following information: We need to find the 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 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