{"id":231643,"date":"2025-06-11T09:21:52","date_gmt":"2025-06-11T09:21:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=231643"},"modified":"2025-06-11T09:21:54","modified_gmt":"2025-06-11T09:21:54","slug":"fill-in-the-blank-to-complete-the-trigonometric-formula-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/fill-in-the-blank-to-complete-the-trigonometric-formula-2\/","title":{"rendered":"Fill in the blank to complete the trigonometric formula"},"content":{"rendered":"\n

Fill in the blank to complete the trigonometric formula. (Select all that apply.) sin u 2 = __<\/strong><\/p>\n\n\n\n

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

To complete the trigonometric identity, we analyze the expression:<\/p>\n\n\n\n

sin(u\/2) = ______<\/strong><\/h3>\n\n\n\n

This refers to the half-angle identity<\/strong> in trigonometry. The correct formula is:sin\u2061(u2)=\u00b11\u2212cos\u2061u2\\sin\\left(\\frac{u}{2}\\right) = \\pm \\sqrt{\\frac{1 – \\cos u}{2}}sin(2u\u200b)=\u00b121\u2212cosu\u200b\u200b<\/p>\n\n\n\n

\n\n\n\n

\u2705 Correct answers:<\/strong><\/h3>\n\n\n\n
    \n
  • sin\u2061(u2)=\u00b11\u2212cos\u2061u2\\sin\\left(\\frac{u}{2}\\right) = \\pm \\sqrt{\\frac{1 – \\cos u}{2}}sin(2u\u200b)=\u00b121\u2212cosu\u200b\u200b<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    \ud83d\udcd8 Explanation (300 words):<\/strong><\/h3>\n\n\n\n

    The half-angle identities are derived from the cosine and sine double-angle identities and are useful for expressing trigonometric functions of half angles (e.g., u2\\frac{u}{2}2u\u200b) in terms of the full angle uuu. In particular, the half-angle identity for sine is:sin\u2061(u2)=\u00b11\u2212cos\u2061u2\\sin\\left(\\frac{u}{2}\\right) = \\pm \\sqrt{\\frac{1 – \\cos u}{2}}sin(2u\u200b)=\u00b121\u2212cosu\u200b\u200b<\/p>\n\n\n\n

    This identity is obtained by rearranging the cosine double-angle identity:cos\u2061(2\u03b8)=1\u22122sin\u20612(\u03b8)\\cos(2\\theta) = 1 – 2\\sin^2(\\theta)cos(2\u03b8)=1\u22122sin2(\u03b8)<\/p>\n\n\n\n

    Solving for sin\u2061(\u03b8)\\sin(\\theta)sin(\u03b8), we get:sin\u20612(\u03b8)=1\u2212cos\u2061(2\u03b8)2\\sin^2(\\theta) = \\frac{1 – \\cos(2\\theta)}{2}sin2(\u03b8)=21\u2212cos(2\u03b8)\u200b<\/p>\n\n\n\n

    Then, replacing \u03b8\\theta\u03b8 with u2\\frac{u}{2}2u\u200b, we find:sin\u20612(u2)=1\u2212cos\u2061u2\\sin^2\\left(\\frac{u}{2}\\right) = \\frac{1 – \\cos u}{2}sin2(2u\u200b)=21\u2212cosu\u200b<\/p>\n\n\n\n

    Taking the square root of both sides gives:sin\u2061(u2)=\u00b11\u2212cos\u2061u2\\sin\\left(\\frac{u}{2}\\right) = \\pm \\sqrt{\\frac{1 – \\cos u}{2}}sin(2u\u200b)=\u00b121\u2212cosu\u200b\u200b<\/p>\n\n\n\n

    The \u00b1 (plus-minus)<\/strong> sign depends on the quadrant in which u2\\frac{u}{2}2u\u200b lies. For example:<\/p>\n\n\n\n

      \n
    • If u2\\frac{u}{2}2u\u200b is in Quadrant I or II, then sine is positive.<\/li>\n\n\n\n
    • If u2\\frac{u}{2}2u\u200b is in Quadrant III or IV, then sine is negative.<\/li>\n<\/ul>\n\n\n\n

      This identity is especially useful in solving trigonometric equations, evaluating angles not commonly found on the unit circle, or integrating trigonometric expressions. Understanding this identity provides a powerful tool in both pure and applied mathematics, especially in calculus and physics where angular functions often need to be rewritten in more convenient forms.<\/p>\n\n\n\n

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

      Fill in the blank to complete the trigonometric formula. (Select all that apply.) sin u 2 = __ The Correct Answer and Explanation is: To complete the trigonometric identity, we analyze the expression: sin(u\/2) = ______ This refers to the half-angle identity in trigonometry. The correct formula is:sin\u2061(u2)=\u00b11\u2212cos\u2061u2\\sin\\left(\\frac{u}{2}\\right) = \\pm \\sqrt{\\frac{1 – \\cos u}{2}}sin(2u\u200b)=\u00b121\u2212cosu\u200b\u200b \u2705 […]<\/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-231643","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231643","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=231643"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231643\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231643"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231643"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231643"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}