{"id":199653,"date":"2025-03-12T08:17:34","date_gmt":"2025-03-12T08:17:34","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=199653"},"modified":"2025-03-12T08:17:36","modified_gmt":"2025-03-12T08:17:36","slug":"discuss-why-this-statement-is-incorrect-arcsinsinx-x-for-all-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/12\/discuss-why-this-statement-is-incorrect-arcsinsinx-x-for-all-x\/","title":{"rendered":"Discuss why this statement is incorrect: arcsin(sin(x)) = x for all x"},"content":{"rendered":"\n<p>Discuss why this statement is incorrect: arcsin(sin(x)) = x for all x. (Fill in the blank using interval notation.) This statement is only correct if x is in the interval The range of arcsin(x) is restricted to this interval because sin(x) is over this interval.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>The statement <strong>arcsin(sin(x)) = x for all x<\/strong> is <strong>incorrect<\/strong> because the inverse sine function, <strong>arcsin(x)<\/strong>, only returns values within a specific range. The correct statement is:<\/p>\n\n\n\n<p>[<br>\\arcsin(\\sin(x)) = x \\quad \\text{only if } x \\text{ is in the interval } \\mathbf{[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]}.<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The function <strong>sin(x)<\/strong> is periodic with a period of (2\\pi), meaning it repeats its values infinitely. However, the <strong>arcsin(x)<\/strong> function, also called the inverse sine function, is defined to return values only in its <strong>principal range<\/strong>:<\/p>\n\n\n\n<p>[<br>\\text{Range of } \\arcsin(x) = \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2} \\right].<br>]<\/p>\n\n\n\n<p>This restriction exists because an inverse function must be <strong>one-to-one<\/strong> (bijective), meaning it has exactly one output for each input. Since the sine function is not one-to-one over all real numbers, we must <strong>restrict<\/strong> it to an interval where it is one-to-one. The interval ( \\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2} \\right] ) is chosen because within this range, <strong>sin(x) is increasing and covers all possible sine values<\/strong> from (-1) to (1) exactly once.<\/p>\n\n\n\n<p>If ( x ) is <strong>outside<\/strong> this interval, the equation (\\arcsin(\\sin(x)) = x) does not always hold. For example:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If ( x = \\frac{3\\pi}{4} ), then <strong>sin(x) = \\sin(\\frac{\\pi}{4})<\/strong>.<\/li>\n\n\n\n<li>However, <strong>arcsin(sin(x))<\/strong> returns ( \\frac{\\pi}{4} ), <strong>not<\/strong> ( \\frac{3\\pi}{4} ), since (\\frac{3\\pi}{4}) is outside the principal range.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the function <strong>arcsin(sin(x))<\/strong> only correctly returns ( x ) when ( x ) is <strong>already within<\/strong> ( [-\\frac{\\pi}{2}, \\frac{\\pi}{2}] ).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Discuss why this statement is incorrect: arcsin(sin(x)) = x for all x. (Fill in the blank using interval notation.) This statement is only correct if x is in the interval The range of arcsin(x) is restricted to this interval because sin(x) is over this interval. The correct answer and explanation is : The statement arcsin(sin(x)) [&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-199653","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199653","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=199653"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/199653\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=199653"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=199653"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=199653"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}