{"id":251970,"date":"2025-07-11T04:28:35","date_gmt":"2025-07-11T04:28:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=251970"},"modified":"2025-07-11T04:28:38","modified_gmt":"2025-07-11T04:28:38","slug":"find-the-exact-value-of-the-quotient","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/11\/find-the-exact-value-of-the-quotient\/","title":{"rendered":"Find the exact value of the quotient"},"content":{"rendered":"\n<p>Find the exact value of the quotient.\\ \\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)}\\ The exact value of \\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)} is<\/p>\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 the quotient cos\u2061(\u03c06)sin\u2061(\u03c06)\\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)}sin(6\u03c0\u200b)cos(6\u03c0\u200b)\u200b, we need to recall the exact trigonometric values of cos\u2061(\u03c06)\\cos\\left(\\frac{\\pi}{6}\\right)cos(6\u03c0\u200b) and sin\u2061(\u03c06)\\sin\\left(\\frac{\\pi}{6}\\right)sin(6\u03c0\u200b).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Know the standard trigonometric values<\/h3>\n\n\n\n<p>From the unit circle or standard trigonometric identities, the values of cosine and sine at \u03c06\\frac{\\pi}{6}6\u03c0\u200b (or 30 degrees) are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cos\u2061(\u03c06)=32\\cos\\left(\\frac{\\pi}{6}\\right) = \\frac{\\sqrt{3}}{2}cos(6\u03c0\u200b)=23\u200b\u200b<\/li>\n\n\n\n<li>sin\u2061(\u03c06)=12\\sin\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{2}sin(6\u03c0\u200b)=21\u200b<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Set up the quotient<\/h3>\n\n\n\n<p>Now, we can substitute these values into the original expression:cos\u2061(\u03c06)sin\u2061(\u03c06)=3212\\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)} = \\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}sin(6\u03c0\u200b)cos(6\u03c0\u200b)\u200b=21\u200b23\u200b\u200b\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Simplify the fraction<\/h3>\n\n\n\n<p>To simplify, divide the numerator by the denominator. Dividing by a fraction is the same as multiplying by its reciprocal. So:3212=32\u00d721=3\\frac{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} = \\frac{\\sqrt{3}}{2} \\times \\frac{2}{1} = \\sqrt{3}21\u200b23\u200b\u200b\u200b=23\u200b\u200b\u00d712\u200b=3\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>Thus, the exact value of the quotient is:3\\boxed{\\sqrt{3}}3\u200b\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The key steps to solving this problem are recognizing the exact values of cos\u2061\\coscos and sin\u2061\\sinsin at \u03c06\\frac{\\pi}{6}6\u03c0\u200b, and then simplifying the fraction by multiplying by the reciprocal of the denominator. This technique is useful for simplifying trigonometric expressions quickly.<\/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-banner6-200.jpeg\" alt=\"\" class=\"wp-image-251971\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of the quotient.\\ \\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)}\\ The exact value of \\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)} is The Correct Answer and Explanation is: To find the exact value of the quotient cos\u2061(\u03c06)sin\u2061(\u03c06)\\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)}sin(6\u03c0\u200b)cos(6\u03c0\u200b)\u200b, we need to recall the exact trigonometric values of cos\u2061(\u03c06)\\cos\\left(\\frac{\\pi}{6}\\right)cos(6\u03c0\u200b) and sin\u2061(\u03c06)\\sin\\left(\\frac{\\pi}{6}\\right)sin(6\u03c0\u200b). Step 1: Know the standard trigonometric values From the unit circle or standard trigonometric [&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-251970","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251970","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=251970"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251970\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=251970"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=251970"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=251970"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}