{"id":217611,"date":"2025-05-22T08:07:49","date_gmt":"2025-05-22T08:07:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217611"},"modified":"2025-05-22T08:07:51","modified_gmt":"2025-05-22T08:07:51","slug":"find-the-exact-value-of-the-following-trigonometric-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/find-the-exact-value-of-the-following-trigonometric-function\/","title":{"rendered":"Find the exact value of the following trigonometric function."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Find the exact value of the following trigonometric function.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">sin 2pi\/3<br>tan 5pi\/6<br>csc 7pi\/4<br>cos 5pi\/3<br>csc 3pi\/4<br>cot pi\/6<br>sec 5pi\/4<br>tan pi\/2<br>If sin theta = 4\/5 and theta is acute, find the exact value of the other five trigonometric functions. No decimals!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If sin theta = -Squareroot 3\/2 and 3 pi\/2 lessthanorequalto theta lessthanorequalto, find cos theta.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><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 class=\"wp-block-paragraph\">Let&#8217;s solve each of the trigonometric expressions <strong>and<\/strong> provide a <strong>300-word explanation<\/strong> for the final question.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Exact Values of Trigonometric Functions:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>sin(2\u03c0\/3):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2\u03c0\/3 is in Quadrant II<\/li>\n\n\n\n<li>Reference angle = \u03c0\/3<\/li>\n\n\n\n<li>sin(\u03c0\/3) = \u221a3\/2 \u2192 In Quadrant II, sine is <strong>positive<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u221a3\/2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>tan(5\u03c0\/6):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5\u03c0\/6 is in Quadrant II<\/li>\n\n\n\n<li>Reference angle = \u03c0\/6<\/li>\n\n\n\n<li>tan(\u03c0\/6) = 1\/\u221a3 \u2192 In Quadrant II, tangent is <strong>negative<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u20131\/\u221a3 or \u2013\u221a3\/3<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>csc(7\u03c0\/4):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>7\u03c0\/4 is in Quadrant IV<\/li>\n\n\n\n<li>Reference angle = \u03c0\/4<\/li>\n\n\n\n<li>sin(\u03c0\/4) = \u221a2\/2 \u2192 csc = 1\/sin \u2192 In Quadrant IV, sine is <strong>negative<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u2013\u221a2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li><strong>cos(5\u03c0\/3):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5\u03c0\/3 is in Quadrant IV<\/li>\n\n\n\n<li>Reference angle = \u03c0\/3<\/li>\n\n\n\n<li>cos(\u03c0\/3) = 1\/2 \u2192 In Quadrant IV, cosine is <strong>positive<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: 1\/2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"5\" class=\"wp-block-list\">\n<li><strong>csc(3\u03c0\/4):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>3\u03c0\/4 is in Quadrant II<\/li>\n\n\n\n<li>Reference angle = \u03c0\/4<\/li>\n\n\n\n<li>sin(\u03c0\/4) = \u221a2\/2 \u2192 csc = 1\/sin<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u221a2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"6\" class=\"wp-block-list\">\n<li><strong>cot(\u03c0\/6):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cot = 1\/tan<\/li>\n\n\n\n<li>tan(\u03c0\/6) = 1\/\u221a3<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u221a3<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"7\" class=\"wp-block-list\">\n<li><strong>sec(5\u03c0\/4):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5\u03c0\/4 is in Quadrant III<\/li>\n\n\n\n<li>Reference angle = \u03c0\/4<\/li>\n\n\n\n<li>cos(\u03c0\/4) = \u221a2\/2 \u2192 In Quadrant III, cosine is <strong>negative<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: \u2013\u221a2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<ol start=\"8\" class=\"wp-block-list\">\n<li><strong>tan(\u03c0\/2):<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>tan(\u03c0\/2) is <strong>undefined<\/strong> because cos(\u03c0\/2) = 0, and tan = sin\/cos<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer: Undefined<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Given: <strong>sin(\u03b8) = 4\/5<\/strong>, \u03b8 is <strong>acute<\/strong>.<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Use Pythagorean identity: sin\u20612(\u03b8)+cos\u20612(\u03b8)=1\u21d2(4\/5)2+cos\u20612(\u03b8)=1\u21d216\/25+cos\u20612(\u03b8)=1\u21d2cos\u20612(\u03b8)=9\/25\u21d2cos\u2061(\u03b8)=3\/5\\sin^2(\u03b8) + \\cos^2(\u03b8) = 1 \\Rightarrow (4\/5)^2 + \\cos^2(\u03b8) = 1 \\Rightarrow 16\/25 + \\cos^2(\u03b8) = 1 \\Rightarrow \\cos^2(\u03b8) = 9\/25 \\Rightarrow \\cos(\u03b8) = 3\/5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u03b8 is acute, so cosine is positive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now find all 6 functions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>sin(\u03b8) = 4\/5<\/strong><\/li>\n\n\n\n<li><strong>cos(\u03b8) = 3\/5<\/strong><\/li>\n\n\n\n<li><strong>tan(\u03b8) = sin\/cos = 4\/3<\/strong><\/li>\n\n\n\n<li><strong>csc(\u03b8) = 1\/sin = 5\/4<\/strong><\/li>\n\n\n\n<li><strong>sec(\u03b8) = 1\/cos = 5\/3<\/strong><\/li>\n\n\n\n<li><strong>cot(\u03b8) = cos\/sin = 3\/4<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Now:<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>If sin(\u03b8) = \u2013\u221a3\/2<\/strong> and <strong>3\u03c0\/2 \u2264 \u03b8 \u2264 2\u03c0<\/strong>, find <strong>cos(\u03b8)<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Explanation (300+ words):<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are told that <strong>sin(\u03b8) = \u2013\u221a3\/2<\/strong> and that <strong>\u03b8 lies between 3\u03c0\/2 and 2\u03c0<\/strong>, which places \u03b8 in the <strong>fourth quadrant<\/strong> (Quadrant IV). In this quadrant, the <strong>sine<\/strong> function is <strong>negative<\/strong>, which matches the given value. The cosine function, however, is <strong>positive<\/strong> in this quadrant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let\u2019s recall the Pythagorean identity: sin\u20612(\u03b8)+cos\u20612(\u03b8)=1\\sin^2(\u03b8) + \\cos^2(\u03b8) = 1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We substitute the known value of sin(\u03b8): (\u22123\/2)2+cos\u20612(\u03b8)=1\u21d23\/4+cos\u20612(\u03b8)=1\u21d2cos\u20612(\u03b8)=1\u22123\/4=1\/4\u21d2cos\u2061(\u03b8)=\u00b11\/2(-\\sqrt{3}\/2)^2 + \\cos^2(\u03b8) = 1 \\Rightarrow 3\/4 + \\cos^2(\u03b8) = 1 \\Rightarrow \\cos^2(\u03b8) = 1 &#8211; 3\/4 = 1\/4 \\Rightarrow \\cos(\u03b8) = \u00b11\/2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now we decide on the correct sign. Since <strong>\u03b8 is in Quadrant IV<\/strong>, and <strong>cosine is positive<\/strong> in this quadrant, we choose the positive root.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Final Answer: cos(\u03b8) = 1\/2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To summarize, understanding which quadrant the angle lies in helps determine the correct sign of the trigonometric function. Even if a value such as cos\u00b2(\u03b8) = 1\/4 leads to both \u00b11\/2, we must choose the sign based on the quadrant. In this case, the negative sine confirms the angle is in the fourth quadrant, and thus cosine must be <strong>positive<\/strong>. This approach ensures we not only compute the correct <strong>m<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>agnitude<\/strong> but also apply corr<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">ect <strong>directional signs<\/strong> according to the unit circle, a fundamental tool in trigonometry.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-18.png\" alt=\"\" class=\"wp-image-217612\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of the following trigonometric function. sin 2pi\/3tan 5pi\/6csc 7pi\/4cos 5pi\/3csc 3pi\/4cot pi\/6sec 5pi\/4tan pi\/2If sin theta = 4\/5 and theta is acute, find the exact value of the other five trigonometric functions. No decimals! If sin theta = -Squareroot 3\/2 and 3 pi\/2 lessthanorequalto theta lessthanorequalto, find cos theta. The Correct [&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-217611","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217611","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=217611"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217611\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217611"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217611"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217611"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}