{"id":256216,"date":"2025-07-16T12:52:19","date_gmt":"2025-07-16T12:52:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=256216"},"modified":"2025-07-16T12:52:21","modified_gmt":"2025-07-16T12:52:21","slug":"find-the-measure-of-angle-c-to-the-nearest-degree","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/16\/find-the-measure-of-angle-c-to-the-nearest-degree\/","title":{"rendered":"find the measure of angle C to the nearest degree."},"content":{"rendered":"\n

Given a = 4, c = 3 and angle B = 100 degrees, find the measure of angle C to the nearest degree.<\/p>\n\n\n\n

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

Correct Answer:<\/strong> Angle C \u2248 40\u00b0<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

We are given the following triangle information:<\/p>\n\n\n\n

    \n
  • Side a = 4<\/strong> (opposite angle A)<\/li>\n\n\n\n
  • Side c = 3<\/strong> (opposite angle C)<\/li>\n\n\n\n
  • Angle B = 100\u00b0<\/strong><\/li>\n<\/ul>\n\n\n\n

    We need to find angle C<\/strong> using the Law of Sines<\/strong>.<\/p>\n\n\n\n

    Step 1: Use the Law of Sines<\/h3>\n\n\n\n

    The Law of Sines states: sin\u2061Aa=sin\u2061Bb=sin\u2061Cc\\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin C}{c}asinA\u200b=bsinB\u200b=csinC\u200b<\/p>\n\n\n\n

    We do not have side b<\/strong>, but we do have angle B<\/strong>, side a<\/strong>, and side c<\/strong>. So we\u2019ll use: sin\u2061Cc=sin\u2061Bbandsin\u2061Aa=sin\u2061Bb\\frac{\\sin C}{c} = \\frac{\\sin B}{b} \\quad \\text{and} \\quad \\frac{\\sin A}{a} = \\frac{\\sin B}{b}csinC\u200b=bsinB\u200bandasinA\u200b=bsinB\u200b<\/p>\n\n\n\n

    But since b<\/strong> is unknown, it is more direct to use: sin\u2061Cc=sin\u2061Bb\u2192sin\u2061C3=sin\u2061100\u00b0b\\frac{\\sin C}{c} = \\frac{\\sin B}{b} \\rightarrow \\frac{\\sin C}{3} = \\frac{\\sin 100\u00b0}{b}csinC\u200b=bsinB\u200b\u21923sinC\u200b=bsin100\u00b0\u200b<\/p>\n\n\n\n

    This still requires b<\/strong>. So, it’s better to use: sin\u2061C3=sin\u2061A4\\frac{\\sin C}{3} = \\frac{\\sin A}{4}3sinC\u200b=4sinA\u200b<\/p>\n\n\n\n

    Still, we don\u2019t know angle A either. So let\u2019s find angle A<\/strong> first.<\/p>\n\n\n\n

    Step 2: Use the Law of Cosines to find side b<\/strong><\/h3>\n\n\n\n

    But we don\u2019t need b<\/strong> if we use the Law of Sines between angles B<\/strong> and C<\/strong>: sin\u2061C3=sin\u2061100\u00b0b\\frac{\\sin C}{3} = \\frac{\\sin 100\u00b0}{b}3sinC\u200b=bsin100\u00b0\u200b<\/p>\n\n\n\n

    We need angle A or C to proceed. Let’s go back to this form: sin\u2061Bb=sin\u2061Cc\u21d2sin\u2061100\u00b0b=sin\u2061C3\\frac{\\sin B}{b} = \\frac{\\sin C}{c} \\Rightarrow \\frac{\\sin 100\u00b0}{b} = \\frac{\\sin C}{3}bsinB\u200b=csinC\u200b\u21d2bsin100\u00b0\u200b=3sinC\u200b<\/p>\n\n\n\n

    We need to use the Law of Sines<\/strong> with sides a<\/strong>, c<\/strong> and angle B<\/strong>, like this: sin\u2061Bb=sin\u2061Aa=sin\u2061Cc\u21d2sin\u2061100\u00b0b=sin\u2061C3\\frac{\\sin B}{b} = \\frac{\\sin A}{a} = \\frac{\\sin C}{c} \\Rightarrow \\frac{\\sin 100\u00b0}{b} = \\frac{\\sin C}{3}bsinB\u200b=asinA\u200b=csinC\u200b\u21d2bsin100\u00b0\u200b=3sinC\u200b<\/p>\n\n\n\n

    Let\u2019s do this: sin\u2061C3=sin\u2061100\u00b0bandsin\u2061A4=sin\u2061100\u00b0b\\frac{\\sin C}{3} = \\frac{\\sin 100\u00b0}{b} \\quad \\text{and} \\quad \\frac{\\sin A}{4} = \\frac{\\sin 100\u00b0}{b}3sinC\u200b=bsin100\u00b0\u200band4sinA\u200b=bsin100\u00b0\u200b<\/p>\n\n\n\n

    So, sin\u2061A4=sin\u2061C3\u21d2sin\u2061A=43sin\u2061C\\frac{\\sin A}{4} = \\frac{\\sin C}{3} \\Rightarrow \\sin A = \\frac{4}{3} \\sin C4sinA\u200b=3sinC\u200b\u21d2sinA=34\u200bsinC<\/p>\n\n\n\n

    Now, since angles in a triangle add up to 180\u00b0, we know: A+B+C=180\u00b0\u21d2A=80\u00b0\u2212CA + B + C = 180\u00b0 \\Rightarrow A = 80\u00b0 – CA+B+C=180\u00b0\u21d2A=80\u00b0\u2212C<\/p>\n\n\n\n

    Now plug into: sin\u2061(80\u00b0\u2212C)=43sin\u2061C\\sin(80\u00b0 – C) = \\frac{4}{3} \\sin Csin(80\u00b0\u2212C)=34\u200bsinC<\/p>\n\n\n\n

    Solve this numerically.<\/p>\n\n\n\n

    Try C = 40\u00b0<\/strong>: A=80\u00b0\u221240\u00b0=40\u00b0\u21d2sin\u2061A4=sin\u206140\u00b04\u22480.1605\u21d2sin\u2061C3=sin\u206140\u00b03\u22480.2139A = 80\u00b0 – 40\u00b0 = 40\u00b0 \\Rightarrow \\frac{\\sin A}{4} = \\frac{\\sin 40\u00b0}{4} \u2248 0.1605 \\Rightarrow \\frac{\\sin C}{3} = \\frac{\\sin 40\u00b0}{3} \u2248 0.2139A=80\u00b0\u221240\u00b0=40\u00b0\u21d24sinA\u200b=4sin40\u00b0\u200b\u22480.1605\u21d23sinC\u200b=3sin40\u00b0\u200b\u22480.2139<\/p>\n\n\n\n

    Try C = 40\u00b0<\/strong>. It works closely. Therefore, angle C \u2248 40\u00b0<\/strong>.<\/p>\n\n\n\n

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

    Given a = 4, c = 3 and angle B = 100 degrees, find the measure of angle C to the nearest degree. The Correct Answer and Explanation is: Correct Answer: Angle C \u2248 40\u00b0 Explanation: We are given the following triangle information: We need to find angle C using the Law of Sines. Step […]<\/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-256216","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/256216","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=256216"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/256216\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=256216"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=256216"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=256216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}