{"id":258873,"date":"2025-07-18T14:28:39","date_gmt":"2025-07-18T14:28:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=258873"},"modified":"2025-07-18T14:28:41","modified_gmt":"2025-07-18T14:28:41","slug":"the-period-of-the-function-fx-sinx-cosx-sinx-cosx-is","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/18\/the-period-of-the-function-fx-sinx-cosx-sinx-cosx-is\/","title":{"rendered":"The period of the function f(x) = |sin(x)| + |cos(x)| \/ |sin(x) &#8211; cos(x)| is"},"content":{"rendered":"\n<p>The period of the function f(x) = |sin(x)| + |cos(x)| \/ |sin(x) &#8211; cos(x)| is A. \u00cf\u20ac\/2 B. \u00cf\u20ac\/4 C. \u00cf\u20ac D. 2\u00cf\u20ac<\/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>The given function is: f(x)=\u2223sin\u2061(x)\u2223+\u2223cos\u2061(x)\u2223\u2223sin\u2061(x)\u2212cos\u2061(x)\u2223f(x) = \\frac{| \\sin(x) | + | \\cos(x) |}{| \\sin(x) &#8211; \\cos(x) |}f(x)=\u2223sin(x)\u2212cos(x)\u2223\u2223sin(x)\u2223+\u2223cos(x)\u2223\u200b<\/p>\n\n\n\n<p>To find the period of this function, let&#8217;s first analyze its components. The sine and cosine functions have periods of 2\u03c02\\pi2\u03c0, but due to the absolute value and the structure of the function, the period of f(x)f(x)f(x) could be different.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Break down the function<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2223sin\u2061(x)\u2223|\\sin(x)|\u2223sin(x)\u2223 and \u2223cos\u2061(x)\u2223|\\cos(x)|\u2223cos(x)\u2223 will have a period of \u03c0\\pi\u03c0 since both sin\u2061(x)\\sin(x)sin(x) and cos\u2061(x)\\cos(x)cos(x) repeat every 2\u03c02\\pi2\u03c0, but the absolute value &#8220;flips&#8221; negative values to positive, making each function repeat at half the period.<\/li>\n\n\n\n<li>The denominator, \u2223sin\u2061(x)\u2212cos\u2061(x)\u2223|\\sin(x) &#8211; \\cos(x)|\u2223sin(x)\u2212cos(x)\u2223, needs further inspection. The term sin\u2061(x)\u2212cos\u2061(x)\\sin(x) &#8211; \\cos(x)sin(x)\u2212cos(x) has a period of 2\u03c02\\pi2\u03c0, but when we take the absolute value, the function will repeat every \u03c0\\pi\u03c0.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Find the combined period<\/h3>\n\n\n\n<p>Since both the numerator and the denominator involve periodic functions with period \u03c0\\pi\u03c0, the overall function will repeat with the same period as the shortest period involved.<\/p>\n\n\n\n<p>Thus, the period of the function f(x)f(x)f(x) is \u03c0\\pi\u03c0.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>The correct answer is <strong>C. \u03c0\\pi\u03c0<\/strong>.<\/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-banner9-257.jpeg\" alt=\"\" class=\"wp-image-258874\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The period of the function f(x) = |sin(x)| + |cos(x)| \/ |sin(x) &#8211; cos(x)| is A. \u00cf\u20ac\/2 B. \u00cf\u20ac\/4 C. \u00cf\u20ac D. 2\u00cf\u20ac The Correct Answer and Explanation is: The given function is: f(x)=\u2223sin\u2061(x)\u2223+\u2223cos\u2061(x)\u2223\u2223sin\u2061(x)\u2212cos\u2061(x)\u2223f(x) = \\frac{| \\sin(x) | + | \\cos(x) |}{| \\sin(x) &#8211; \\cos(x) |}f(x)=\u2223sin(x)\u2212cos(x)\u2223\u2223sin(x)\u2223+\u2223cos(x)\u2223\u200b To find the period of this function, let&#8217;s first [&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-258873","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258873","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=258873"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/258873\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=258873"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=258873"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=258873"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}