{"id":199244,"date":"2025-03-10T20:48:28","date_gmt":"2025-03-10T20:48:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=199244"},"modified":"2025-03-10T20:48:31","modified_gmt":"2025-03-10T20:48:31","slug":"find-the-exact-value-of-each-trigonometric-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/10\/find-the-exact-value-of-each-trigonometric-function\/","title":{"rendered":"Find the exact value of each trigonometric function"},"content":{"rendered":"\n<p>Find the exact value of each trigonometric function<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>COS 135 degrees<\/li>\n\n\n\n<li>TAN 930 degrees<\/li>\n\n\n\n<li>COT 810 degrees\u00c3\u201a\u00c2<\/li>\n\n\n\n<li>sec -240 degrees<\/li>\n<\/ol>\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>Let&#8217;s go step by step to find the exact values of the given trigonometric functions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. ( \\cos 135^\\circ )<\/strong><\/h3>\n\n\n\n<p>The angle ( 135^\\circ ) is in the second quadrant. The reference angle is:<br>[<br>180^\\circ &#8211; 135^\\circ = 45^\\circ<br>]<br>Since cosine is negative in the second quadrant:<br>[<br>\\cos 135^\\circ = -\\cos 45^\\circ = -\\frac{\\sqrt{2}}{2}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. ( \\tan 930^\\circ )<\/strong><\/h3>\n\n\n\n<p>Since tangent has a period of ( 180^\\circ ), we reduce ( 930^\\circ ) by subtracting multiples of ( 360^\\circ ):<\/p>\n\n\n\n<p>[<br>930^\\circ &#8211; 2(360^\\circ) = 930^\\circ &#8211; 720^\\circ = 210^\\circ<br>]<\/p>\n\n\n\n<p>The reference angle is:<\/p>\n\n\n\n<p>[<br>210^\\circ &#8211; 180^\\circ = 30^\\circ<br>]<\/p>\n\n\n\n<p>Since tangent is positive in the third quadrant:<\/p>\n\n\n\n<p>[<br>\\tan 210^\\circ = \\tan 30^\\circ = \\frac{1}{\\sqrt{3}} = -\\frac{\\sqrt{3}}{3}<br>]<\/p>\n\n\n\n<p>Thus,<\/p>\n\n\n\n<p>[<br>\\tan 930^\\circ = -\\frac{\\sqrt{3}}{3}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>3. ( \\cot 810^\\circ )<\/strong><\/h3>\n\n\n\n<p>Since cotangent has a period of ( 180^\\circ ), we reduce ( 810^\\circ ) by subtracting multiples of ( 180^\\circ ):<\/p>\n\n\n\n<p>[<br>810^\\circ &#8211; 4(180^\\circ) = 810^\\circ &#8211; 720^\\circ = 90^\\circ<br>]<\/p>\n\n\n\n<p>Since ( \\cot 90^\\circ = 0 ), we conclude:<\/p>\n\n\n\n<p>[<br>\\cot 810^\\circ = 0<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>4. ( \\sec (-240^\\circ) )<\/strong><\/h3>\n\n\n\n<p>Since secant is the reciprocal of cosine, we first find ( \\cos (-240^\\circ) ). Adding ( 360^\\circ ) to bring it within a standard range:<\/p>\n\n\n\n<p>[<br>-240^\\circ + 360^\\circ = 120^\\circ<br>]<\/p>\n\n\n\n<p>The reference angle is:<\/p>\n\n\n\n<p>[<br>180^\\circ &#8211; 120^\\circ = 60^\\circ<br>]<\/p>\n\n\n\n<p>Since cosine is negative in the second quadrant:<\/p>\n\n\n\n<p>[<br>\\cos 120^\\circ = -\\cos 60^\\circ = -\\frac{1}{2}<br>]<\/p>\n\n\n\n<p>Now, taking the reciprocal:<\/p>\n\n\n\n<p>[<br>\\sec (-240^\\circ) = \\sec 120^\\circ = \\frac{1}{\\cos 120^\\circ} = \\frac{1}{-1\/2} = -2<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final Answers:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>( \\cos 135^\\circ = -\\frac{\\sqrt{2}}{2} )<\/li>\n\n\n\n<li>( \\tan 930^\\circ = -\\frac{\\sqrt{3}}{3} )<\/li>\n\n\n\n<li>( \\cot 810^\\circ = 0 )<\/li>\n\n\n\n<li>( \\sec (-240^\\circ) = -2 )<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300 words)<\/strong><\/h3>\n\n\n\n<p>To solve these problems, we used fundamental trigonometric concepts, including reference angles, quadrant rules, and periodicity.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>For ( \\cos 135^\\circ )<\/strong>, we recognized that ( 135^\\circ ) lies in the second quadrant where cosine is negative. Using the reference angle ( 45^\\circ ), we found ( \\cos 135^\\circ = -\\frac{\\sqrt{2}}{2} ).<\/li>\n\n\n\n<li><strong>For ( \\tan 930^\\circ )<\/strong>, we reduced the angle by subtracting multiples of ( 360^\\circ ) to bring it within a standard range, leading to ( 210^\\circ ). Since ( 210^\\circ ) is in the third quadrant, where tangent is positive, we found ( \\tan 210^\\circ = -\\frac{\\sqrt{3}}{3} ).<\/li>\n\n\n\n<li><strong>For ( \\cot 810^\\circ )<\/strong>, we reduced the angle using the periodicity of ( 180^\\circ ), bringing it to ( 90^\\circ ). Since ( \\cot 90^\\circ = 0 ), the final answer was 0.<\/li>\n\n\n\n<li><strong>For ( \\sec (-240^\\circ) )<\/strong>, we converted the negative angle to a positive equivalent by adding ( 360^\\circ ), leading to ( 120^\\circ ). We found ( \\cos 120^\\circ = -\\frac{1}{2} ) and took its reciprocal to get ( \\sec 120^\\circ = -2 ).<\/li>\n<\/ol>\n\n\n\n<p>Understanding quadrant signs, reference angles, and periodic properties of trigonometric functions allows us to determine exact values efficiently.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of each trigonometric function The correct answer and explanation is : Let&#8217;s go step by step to find the exact values of the given trigonometric functions. 1. ( \\cos 135^\\circ ) The angle ( 135^\\circ ) is in the second quadrant. The reference angle is:[180^\\circ &#8211; 135^\\circ = 45^\\circ]Since cosine is [&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-199244","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199244","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=199244"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199244\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=199244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=199244"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=199244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}