To calculate the exact value of cot\u2061(13\u03c06)\\cot\\left(\\frac{13\\pi}{6}\\right)cot(613\u03c0\u200b), begin by expressing the angle in terms of a coterminal angle within [0,2\u03c0][0, 2\\pi][0,2\u03c0].<\/p>\n\n\n\n
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Step 1: Find a coterminal angle between 000 and 2\u03c02\\pi2\u03c0<\/strong>13\u03c06\u22122\u03c0=13\u03c06\u221212\u03c06=\u03c06\\frac{13\\pi}{6} – 2\\pi = \\frac{13\\pi}{6} – \\frac{12\\pi}{6} = \\frac{\\pi}{6}613\u03c0\u200b\u22122\u03c0=613\u03c0\u200b\u2212612\u03c0\u200b=6\u03c0\u200b<\/p>\n\n\n\n Step 2: Use the coterminal identity<\/strong>cot\u2061(13\u03c06)=cot\u2061(\u03c06)\\cot\\left(\\frac{13\\pi}{6}\\right) = \\cot\\left(\\frac{\\pi}{6}\\right)cot(613\u03c0\u200b)=cot(6\u03c0\u200b)<\/p>\n\n\n\n Step 3: Use the definition of cotangent in terms of sine and cosine<\/strong>cot\u2061(\u03c06)=cos\u2061(\u03c06)sin\u2061(\u03c06)\\cot\\left(\\frac{\\pi}{6}\\right) = \\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)}cot(6\u03c0\u200b)=sin(6\u03c0\u200b)cos(6\u03c0\u200b)\u200b<\/p>\n\n\n\n Step 4: Use known exact values<\/strong>cos\u2061(\u03c06)=32,sin\u2061(\u03c06)=12\\cos\\left(\\frac{\\pi}{6}\\right) = \\frac{\\sqrt{3}}{2}, \\quad \\sin\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{2}cos(6\u03c0\u200b)=23\u200b\u200b,sin(6\u03c0\u200b)=21\u200bcot\u2061(\u03c06)=3\/21\/2=3\\cot\\left(\\frac{\\pi}{6}\\right) = \\frac{\\sqrt{3}\/2}{1\/2} = \\sqrt{3}cot(6\u03c0\u200b)=1\/23\u200b\/2\u200b=3\u200b<\/p>\n\n\n\n Final Answer:<\/strong>cot\u2061(13\u03c06)=3\\cot\\left(\\frac{13\\pi}{6}\\right) = \\sqrt{3}cot(613\u03c0\u200b)=3\u200b<\/p>\n\n\n\n Explanation<\/strong> The cotangent function, defined as the ratio of cosine to sine, leads to a straightforward computation. The unit circle values for cos\u2061(\u03c06)\\cos\\left(\\frac{\\pi}{6}\\right)cos(6\u03c0\u200b) and sin\u2061(\u03c06)\\sin\\left(\\frac{\\pi}{6}\\right)sin(6\u03c0\u200b) are commonly memorized or referenced from standard trigonometric tables. Specifically, the cosine of \u03c06\\frac{\\pi}{6}6\u03c0\u200b is 32\\frac{\\sqrt{3}}{2}23\u200b\u200b, and the sine is 12\\frac{1}{2}21\u200b. Dividing these values gives:cos\u2061(\u03c06)sin\u2061(\u03c06)=3\/21\/2=3\\frac{\\cos\\left(\\frac{\\pi}{6}\\right)}{\\sin\\left(\\frac{\\pi}{6}\\right)} = \\frac{\\sqrt{3}\/2}{1\/2} = \\sqrt{3}sin(6\u03c0\u200b)cos(6\u03c0\u200b)\u200b=1\/23\u200b\/2\u200b=3\u200b<\/p>\n\n\n\n Thus, the exact value of cot\u2061(13\u03c06)\\cot\\left(\\frac{13\\pi}{6}\\right)cot(613\u03c0\u200b) is 3\\sqrt{3}3\u200b. This result showcases the usefulness of understanding coterminal angles and fundamental trigonometric identities, allowing for efficient and accurate evaluations of trigonometric expressions.<\/p>\n\n\n\n Use trigonometric identities and compound angle formulas to calculate the exact value of \\cot\\left(\\frac{13\\pi}{6}\\right). Show at least four lines of work for full marks. [4 Marks] The Correct Answer and Explanation is: To calculate the exact value of cot\u2061(13\u03c06)\\cot\\left(\\frac{13\\pi}{6}\\right)cot(613\u03c0\u200b), begin by expressing the angle in terms of a coterminal angle within [0,2\u03c0][0, 2\\pi][0,2\u03c0]. Step 1: […]<\/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-235557","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235557","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=235557"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/235557\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=235557"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=235557"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=235557"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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The cotangent of an angle that exceeds 2\u03c02\\pi2\u03c0 can be simplified by identifying a coterminal angle. A coterminal angle shares the same terminal side on the unit circle as the original angle. To find it, subtract 2\u03c02\\pi2\u03c0 from 13\u03c06\\frac{13\\pi}{6}613\u03c0\u200b. This operation yields \u03c06\\frac{\\pi}{6}6\u03c0\u200b, which lies within the principal interval from 000 to 2\u03c02\\pi2\u03c0. This implies that the trigonometric value of cot\u2061(13\u03c06)\\cot\\left(\\frac{13\\pi}{6}\\right)cot(613\u03c0\u200b) is identical to that of cot\u2061(\u03c06)\\cot\\left(\\frac{\\pi}{6}\\right)cot(6\u03c0\u200b).<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-599.jpeg\")
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