{"id":159350,"date":"2024-10-31T16:44:32","date_gmt":"2024-10-31T16:44:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=159350"},"modified":"2024-10-31T16:44:35","modified_gmt":"2024-10-31T16:44:35","slug":"explain-how-1-radian-is-equal-to-1-2%cf%80-revolutions","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/31\/explain-how-1-radian-is-equal-to-1-2%cf%80-revolutions\/","title":{"rendered":"Explain how 1 radian is equal to 1\/2\u03c0 revolutions."},"content":{"rendered":"\n<p>Explain how 1 radian is equal to 1\/2\u03c0 revolutions.<\/p>\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 understand how 1 radian is equal to ( \\frac{1}{2\\pi} ) revolutions, we first need to define both radians and revolutions in the context of circular motion.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Definitions:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Radian<\/strong>: A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Therefore, if you have a circle of radius ( r ), an angle of 1 radian corresponds to an arc length of ( r ).<\/li>\n\n\n\n<li><strong>Revolution<\/strong>: A full revolution around a circle refers to a complete rotation of 360 degrees, which is equivalent to ( 2\\pi ) radians. Thus, one revolution corresponds to an angle of ( 2\\pi ) radians.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Derivation:<\/h3>\n\n\n\n<p>To find the relationship between radians and revolutions, we can set up the following equivalence based on the definitions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since one revolution corresponds to ( 2\\pi ) radians, we can say:<br>[<br>1 \\text{ revolution} = 2\\pi \\text{ radians}<br>]<\/li>\n<\/ul>\n\n\n\n<p>To express 1 radian in terms of revolutions, we can rearrange the equation:<br>[<br>1 \\text{ radian} = \\frac{1}{2\\pi} \\text{ revolutions}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This relationship emerges from the definition of the circle itself. The circumference of a circle, which is the total distance around the circle, is given by the formula ( C = 2\\pi r ), where ( r ) is the radius. When we consider the unit circle (a circle with a radius of 1), the circumference becomes ( 2\\pi ).<\/p>\n\n\n\n<p>When the arc length equals the radius (1 unit), we have defined 1 radian. Since the full circumference of the unit circle represents one complete revolution (or ( 2\\pi ) radians), we conclude that the angle of 1 radian is simply the fraction of the total circumference represented by that arc length. Therefore, dividing the total number of radians in a revolution by ( 2\\pi ) gives us the fraction of a revolution that corresponds to 1 radian, leading us to the conclusion that:<\/p>\n\n\n\n<p>[<br>1 \\text{ radian} = \\frac{1}{2\\pi} \\text{ revolutions}<br>]<\/p>\n\n\n\n<p>This mathematical relationship illustrates the connection between linear distance traveled along a circle and the angular measurement in radians and revolutions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Explain how 1 radian is equal to 1\/2\u03c0 revolutions. The Correct Answer and Explanation is: To understand how 1 radian is equal to ( \\frac{1}{2\\pi} ) revolutions, we first need to define both radians and revolutions in the context of circular motion. Definitions: Derivation: To find the relationship between radians and revolutions, we can set [&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-159350","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159350","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=159350"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159350\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=159350"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=159350"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=159350"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}