{"id":238583,"date":"2025-06-18T08:43:42","date_gmt":"2025-06-18T08:43:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238583"},"modified":"2025-06-18T08:43:44","modified_gmt":"2025-06-18T08:43:44","slug":"simplify-sin270-degrees-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/18\/simplify-sin270-degrees-x\/","title":{"rendered":"Simplify sin(270 degrees – x)."},"content":{"rendered":"\n

Simplify sin(270 degrees – x).<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Final Answer:<\/h3>\n\n\n\n

sin\u2061(270\u2218\u2212x)=\u2212cos\u2061(x)\\sin(270^\\circ – x) = -\\cos(x)sin(270\u2218\u2212x)=\u2212cos(x)<\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To simplify sin\u2061(270\u2218\u2212x)\\sin(270^\\circ – x)sin(270\u2218\u2212x), we use a trigonometric identity<\/strong> known as the angle difference identity for sine<\/strong>:sin\u2061(A\u2212B)=sin\u2061Acos\u2061B\u2212cos\u2061Asin\u2061B\\sin(A – B) = \\sin A \\cos B – \\cos A \\sin Bsin(A\u2212B)=sinAcosB\u2212cosAsinB<\/p>\n\n\n\n

Let us apply this to the given expression:sin\u2061(270\u2218\u2212x)=sin\u2061(270\u2218)cos\u2061(x)\u2212cos\u2061(270\u2218)sin\u2061(x)\\sin(270^\\circ – x) = \\sin(270^\\circ)\\cos(x) – \\cos(270^\\circ)\\sin(x)sin(270\u2218\u2212x)=sin(270\u2218)cos(x)\u2212cos(270\u2218)sin(x)<\/p>\n\n\n\n

We now recall the exact trigonometric values<\/strong> for 270\u2218270^\\circ270\u2218:<\/p>\n\n\n\n

    \n
  • sin\u2061(270\u2218)=\u22121\\sin(270^\\circ) = -1sin(270\u2218)=\u22121<\/li>\n\n\n\n
  • cos\u2061(270\u2218)=0\\cos(270^\\circ) = 0cos(270\u2218)=0<\/li>\n<\/ul>\n\n\n\n

    Substitute these values into the identity:sin\u2061(270\u2218\u2212x)=(\u22121)cos\u2061(x)\u2212(0)sin\u2061(x)\\sin(270^\\circ – x) = (-1)\\cos(x) – (0)\\sin(x)sin(270\u2218\u2212x)=(\u22121)cos(x)\u2212(0)sin(x)=\u2212cos\u2061(x)= -\\cos(x)=\u2212cos(x)<\/p>\n\n\n\n

    So, the simplified form is:\u2212cos\u2061(x)\\boxed{-\\cos(x)}\u2212cos(x)\u200b<\/p>\n\n\n\n

    \n\n\n\n

    Geometric Understanding:<\/h3>\n\n\n\n

    The unit circle can also help us visualize this result. The angle 270\u2218270^\\circ270\u2218 lies on the negative y-axis, pointing straight downward. When you subtract an angle xxx from 270\u2218270^\\circ270\u2218, you are rotating clockwise<\/strong> from the downward position by xxx degrees.<\/p>\n\n\n\n

    This brings you into the third quadrant<\/strong> of the unit circle, where sine is negative and cosine is also negative. The reference angle becomes xxx, and the identity shows that the sine of this angle is the negative cosine of xxx, which aligns with the algebraic result we derived.<\/p>\n\n\n\n

    \n\n\n\n

    Summary:<\/h3>\n\n\n\n

    Using identities and values from the unit circle, we showed that:sin\u2061(270\u2218\u2212x)=\u2212cos\u2061(x)\\sin(270^\\circ – x) = -\\cos(x)sin(270\u2218\u2212x)=\u2212cos(x)<\/p>\n\n\n\n

    This is an example of how trigonometric expressions involving angle differences can be simplified using fundamental identities and known angle values.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Simplify sin(270 degrees – x). The Correct Answer and Explanation is: Final Answer: sin\u2061(270\u2218\u2212x)=\u2212cos\u2061(x)\\sin(270^\\circ – x) = -\\cos(x)sin(270\u2218\u2212x)=\u2212cos(x) Explanation: To simplify sin\u2061(270\u2218\u2212x)\\sin(270^\\circ – x)sin(270\u2218\u2212x), we use a trigonometric identity known as the angle difference identity for sine:sin\u2061(A\u2212B)=sin\u2061Acos\u2061B\u2212cos\u2061Asin\u2061B\\sin(A – B) = \\sin A \\cos B – \\cos A \\sin Bsin(A\u2212B)=sinAcosB\u2212cosAsinB Let us apply this to the […]<\/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-238583","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238583","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=238583"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238583\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238583"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238583"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}