{"id":233676,"date":"2025-06-13T10:05:30","date_gmt":"2025-06-13T10:05:30","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=233676"},"modified":"2025-06-13T10:05:32","modified_gmt":"2025-06-13T10:05:32","slug":"what-is-the-exact-value-of-cos-30","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/13\/what-is-the-exact-value-of-cos-30\/","title":{"rendered":"What is the exact value of cos 30"},"content":{"rendered":"\n<p>What is the exact value of cos 30\u00b0 ? Enter your answer, as a simplified fraction, in the box.<\/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><strong>Correct Answer:<\/strong> cos\u206130\u2218=32\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}cos30\u2218=23\u200b\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>To find the exact value of cos\u206130\u2218\\cos 30^\\circcos30\u2218, it&#8217;s helpful to recall the special angles in trigonometry and their corresponding values on the unit circle or within special right triangles.<\/p>\n\n\n\n<p>One common way to find this value is by using the <strong>30\u00b0-60\u00b0-90\u00b0 special right triangle<\/strong>. This type of triangle has angles of 30\u00b0, 60\u00b0, and 90\u00b0, and side lengths in a consistent ratio:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The side opposite the 30\u00b0 angle is <strong>1<\/strong>.<\/li>\n\n\n\n<li>The side opposite the 60\u00b0 angle is <strong>3\\sqrt{3}3\u200b<\/strong>.<\/li>\n\n\n\n<li>The hypotenuse is <strong>2<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>This triangle can be derived by cutting an equilateral triangle (where all sides are equal and all angles are 60\u00b0) in half. When cut down the middle, two 30\u00b0-60\u00b0-90\u00b0 triangles are formed.<\/p>\n\n\n\n<p>In trigonometry, cosine of an angle in a right triangle is defined as the <strong>adjacent side divided by the hypotenuse<\/strong>. Applying this definition to the 30\u00b0 angle in the triangle:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The side adjacent to the 30\u00b0 angle is 3\\sqrt{3}3\u200b.<\/li>\n\n\n\n<li>The hypotenuse is 2.<\/li>\n<\/ul>\n\n\n\n<p>Thus, cos\u206130\u2218=adjacenthypotenuse=32\\cos 30^\\circ = \\frac{\\text{adjacent}}{\\text{hypotenuse}} = \\frac{\\sqrt{3}}{2}cos30\u2218=hypotenuseadjacent\u200b=23\u200b\u200b<\/p>\n\n\n\n<p>This value is exact and commonly used in both pure and applied mathematics. It appears often in geometry, physics, engineering, and trigonometric identities.<\/p>\n\n\n\n<p>It\u2019s also confirmed using the unit circle, where the coordinates of a point corresponding to an angle represent (cos\u2061\u03b8,sin\u2061\u03b8)(\\cos \\theta, \\sin \\theta)(cos\u03b8,sin\u03b8). For 30\u221830^\\circ30\u2218 or \u03c06\\frac{\\pi}{6}6\u03c0\u200b radians, the coordinates are (32,12)\\left( \\frac{\\sqrt{3}}{2}, \\frac{1}{2} \\right)(23\u200b\u200b,21\u200b), so: cos\u206130\u2218=32\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}cos30\u2218=23\u200b\u200b<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-300.jpeg\" alt=\"\" class=\"wp-image-233677\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>What is the exact value of cos 30\u00b0 ? Enter your answer, as a simplified fraction, in the box. The Correct Answer and Explanation is: Correct Answer: cos\u206130\u2218=32\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}cos30\u2218=23\u200b\u200b Explanation: To find the exact value of cos\u206130\u2218\\cos 30^\\circcos30\u2218, it&#8217;s helpful to recall the special angles in trigonometry and their corresponding values on the [&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-233676","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233676","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=233676"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/233676\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=233676"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=233676"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=233676"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}