{"id":230782,"date":"2025-06-10T06:06:54","date_gmt":"2025-06-10T06:06:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=230782"},"modified":"2025-06-10T06:06:56","modified_gmt":"2025-06-10T06:06:56","slug":"find-the-exact-function-value","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/10\/find-the-exact-function-value\/","title":{"rendered":"Find the exact function value."},"content":{"rendered":"\n

Find the exact function value.\\<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The problem is to find the exact value of<\/strong>tan\u2061(5\u03c06)\\tan\\left(\\frac{5\\pi}{6}\\right)tan(65\u03c0\u200b)<\/p>\n\n\n\n

\n\n\n\n

\u2705 Correct Answer:<\/strong><\/h3>\n\n\n\n

tan\u2061(5\u03c06)=\u221213\\tan\\left(\\frac{5\\pi}{6}\\right) = -\\frac{1}{\\sqrt{3}}tan(65\u03c0\u200b)=\u22123\u200b1\u200b<\/p>\n\n\n\n

\n\n\n\n

\u270d\ufe0f Explanation<\/strong><\/h3>\n\n\n\n

To find the exact value of tan\u2061(5\u03c06)\\tan\\left(\\frac{5\\pi}{6}\\right)tan(65\u03c0\u200b), we begin by identifying which quadrant this angle lies in and using a reference angle.<\/p>\n\n\n\n

Step 1: Understand the angle<\/h4>\n\n\n\n

5\u03c06 radians=150\u2218\\frac{5\\pi}{6} \\text{ radians} = 150^\\circ65\u03c0\u200b radians=150\u2218<\/p>\n\n\n\n

This angle lies in Quadrant II<\/strong> (between 90\u221890^\\circ90\u2218 and 180\u2218180^\\circ180\u2218).<\/p>\n\n\n\n

Step 2: Find the reference angle<\/h4>\n\n\n\n

To find the reference angle in Quadrant II, subtract the angle from \u03c0\\pi\u03c0 (or 180\u00b0):Reference angle=\u03c0\u22125\u03c06=\u03c06\\text{Reference angle} = \\pi – \\frac{5\\pi}{6} = \\frac{\\pi}{6}Reference angle=\u03c0\u221265\u03c0\u200b=6\u03c0\u200b<\/p>\n\n\n\n

So, the reference angle is \u03c06\\frac{\\pi}{6}6\u03c0\u200b.<\/p>\n\n\n\n

Step 3: Use known values for tan\u2061(\u03c06)\\tan(\\frac{\\pi}{6})tan(6\u03c0\u200b)<\/h4>\n\n\n\n

From the unit circle, we know:tan\u2061(\u03c06)=13\\tan\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{\\sqrt{3}}tan(6\u03c0\u200b)=3\u200b1\u200b<\/p>\n\n\n\n

Step 4: Determine the sign in Quadrant II<\/h4>\n\n\n\n

In Quadrant II:<\/p>\n\n\n\n

    \n
  • Sine<\/strong> is positive<\/li>\n\n\n\n
  • Cosine<\/strong> is negative<\/li>\n\n\n\n
  • Therefore, Tangent<\/strong> (which is sin\u2061\/cos\u2061\\sin\/\\cossin\/cos) is negative<\/strong><\/li>\n<\/ul>\n\n\n\n

    So:tan\u2061(5\u03c06)=\u2212tan\u2061(\u03c06)=\u221213\\tan\\left(\\frac{5\\pi}{6}\\right) = -\\tan\\left(\\frac{\\pi}{6}\\right) = -\\frac{1}{\\sqrt{3}}tan(65\u03c0\u200b)=\u2212tan(6\u03c0\u200b)=\u22123\u200b1\u200b<\/p>\n\n\n\n

    \n\n\n\n

    \ud83d\udd01 Final Answer Box:<\/h3>\n\n\n\n

    tan\u2061(5\u03c06)=\u221213\\boxed{\\tan\\left(\\frac{5\\pi}{6}\\right) = -\\frac{1}{\\sqrt{3}}}tan(65\u03c0\u200b)=\u22123\u200b1\u200b\u200b<\/p>\n\n\n\n

    This value is exact and comes from understanding the unit circle and reference angles, which is a key part of trigonometry.<\/p>\n\n\n\n

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

    Find the exact function value.\\ The problem is to find the exact value oftan\u2061(5\u03c06)\\tan\\left(\\frac{5\\pi}{6}\\right)tan(65\u03c0\u200b) \u2705 Correct Answer: tan\u2061(5\u03c06)=\u221213\\tan\\left(\\frac{5\\pi}{6}\\right) = -\\frac{1}{\\sqrt{3}}tan(65\u03c0\u200b)=\u22123\u200b1\u200b \u270d\ufe0f Explanation To find the exact value of tan\u2061(5\u03c06)\\tan\\left(\\frac{5\\pi}{6}\\right)tan(65\u03c0\u200b), we begin by identifying which quadrant this angle lies in and using a reference angle. Step 1: Understand the angle 5\u03c06 radians=150\u2218\\frac{5\\pi}{6} \\text{ radians} = 150^\\circ65\u03c0\u200b radians=150\u2218 This […]<\/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-230782","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230782","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=230782"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230782\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=230782"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=230782"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=230782"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}