{"id":221013,"date":"2025-05-29T05:53:20","date_gmt":"2025-05-29T05:53:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221013"},"modified":"2025-05-29T05:53:23","modified_gmt":"2025-05-29T05:53:23","slug":"find-the-reference-number-for-each-value-of-t","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/find-the-reference-number-for-each-value-of-t\/","title":{"rendered":"Find the reference number for each value of t."},"content":{"rendered":"\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-506.png\" alt=\"\" class=\"wp-image-221014\"\/><\/figure>\n\n\n\n<p>Find the terminal point P(x, y) on the unit circle determined by the given value of t. t = 3 pi\/2 Find the reference number for each value of t.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a) t = 13 pi\/6<\/li>\n\n\n\n<li>(b) t = -9pi\/7<\/li>\n\n\n\n<li>(c) t = 6<\/li>\n\n\n\n<li>(d) t = -7<\/li>\n<\/ul>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Terminal Point on the Unit Circle<\/strong><\/h3>\n\n\n\n<p><strong>Given:<\/strong><br>t=3\u03c02t = \\frac{3\\pi}{2}<\/p>\n\n\n\n<p>On the unit circle, the coordinates of a point corresponding to an angle tt (in radians) are: P(x,y)=(cos\u2061t,sin\u2061t)P(x, y) = (\\cos t, \\sin t)<\/p>\n\n\n\n<p>So: cos\u2061(3\u03c02)=0,sin\u2061(3\u03c02)=\u22121\\cos\\left(\\frac{3\\pi}{2}\\right) = 0,\\quad \\sin\\left(\\frac{3\\pi}{2}\\right) = -1<\/p>\n\n\n\n<p><strong>Answer:<\/strong> P(x,y)=(0,\u22121)P(x, y) = (0, -1)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Reference Number for Each Value of tt<\/strong><\/h3>\n\n\n\n<p>A <strong>reference number<\/strong> is the acute angle (positive angle \u2264 \u03c02\\frac{\\pi}{2}) formed by the terminal side of the angle tt and the x-axis. Here&#8217;s how to find it for each value:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(a) t=13\u03c06t = \\frac{13\\pi}{6}<\/strong><\/h4>\n\n\n\n<p>Since this is more than 2\u03c02\\pi, subtract 2\u03c02\\pi: 13\u03c06\u22122\u03c0=13\u03c0\u221212\u03c06=\u03c06\\frac{13\\pi}{6} &#8211; 2\\pi = \\frac{13\\pi &#8211; 12\\pi}{6} = \\frac{\\pi}{6}<\/p>\n\n\n\n<p><strong>Reference number: \u03c06\\frac{\\pi}{6}<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(b) t=\u22129\u03c07t = -\\frac{9\\pi}{7}<\/strong><\/h4>\n\n\n\n<p>Add 2\u03c02\\pi until you get a positive coterminal angle: \u22129\u03c07+2\u03c0=\u22129\u03c07+14\u03c07=5\u03c07-\\frac{9\\pi}{7} + 2\\pi = -\\frac{9\\pi}{7} + \\frac{14\\pi}{7} = \\frac{5\\pi}{7}<\/p>\n\n\n\n<p>This is between 0 and \u03c0\\pi, so the reference angle is just:<\/p>\n\n\n\n<p><strong>Reference number: 5\u03c07\\frac{5\\pi}{7}<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(c) t=6t = 6<\/strong> (in radians)<\/h4>\n\n\n\n<p>Find coterminal angle by subtracting 2\u03c0\u22486.2832\\pi \\approx 6.283: 6\u22122\u03c0\u22486\u22126.283=\u22120.2836 &#8211; 2\\pi \\approx 6 &#8211; 6.283 = -0.283<\/p>\n\n\n\n<p>Then add 2\u03c02\\pi to make it positive: -0.283 + 2\\pi \\approx 6.0 \\) (original value So reference number is: \\[ \\pi &#8211; 6 \\approx 3.142 &#8211; 6 = \\text{Not valid}<\/p>\n\n\n\n<p>Better: find closest multiple of \u03c0\\pi, say t\u2248\u03c0\u22120.283\u21d2t \\approx \\pi &#8211; 0.283 \\Rightarrow reference angle is 2\u03c0\u22126\u22480.2832\\pi &#8211; 6 \\approx 0.283<\/p>\n\n\n\n<p><strong>Reference number: \u22482\u03c0\u22126=0.283\\approx 2\\pi &#8211; 6 = 0.283<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(d) t=\u22127t = -7<\/strong><\/h4>\n\n\n\n<p>Add 2\u03c0\u22486.2832\\pi \\approx 6.283 multiple times to make it positive: \u22127+2\u03c0\u2248\u22127+6.283=\u22120.717\u21d2Still&nbsp;negative-7 + 2\\pi \\approx -7 + 6.283 = -0.717 \\Rightarrow \\text{Still negative}<\/p>\n\n\n\n<p>Add another 2\u03c02\\pi: \u22120.717+6.283=5.566-0.717 + 6.283 = 5.566<\/p>\n\n\n\n<p>Now find the reference number: 2\u03c0\u22125.566\u22486.283\u22125.566=0.7172\\pi &#8211; 5.566 \\approx 6.283 &#8211; 5.566 = 0.717<\/p>\n\n\n\n<p><strong>Reference number: \u22480.717\\approx 0.717<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Answers<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>P(x,y)=(0,\u22121)P(x, y) = (0, -1)<\/li>\n\n\n\n<li>Reference Numbers:\n<ul class=\"wp-block-list\">\n<li>(a) \u03c06\\frac{\\pi}{6}<\/li>\n\n\n\n<li>(b) 5\u03c07\\frac{5\\pi}{7}<\/li>\n\n\n\n<li>(c) \u22480.283\\approx 0.283<\/li>\n\n\n\n<li>(d) \u22480.717\\approx 0.717<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>In trigonometry, angles can be measured in radians, and the unit circle helps us understand their behavior. A unit circle has a radius of 1, and any angle tt drawn in standard position will intersect the circle at a point P(x,y)P(x, y), where x=cos\u2061tx = \\cos t and y=sin\u2061ty = \\sin t. For example, at t=3\u03c02t = \\frac{3\\pi}{2}, the terminal side points straight down along the y-axis, so the point on the unit circle is (0,\u22121)(0, -1).<\/p>\n\n\n\n<p>The reference number is a concept used to simplify angle analysis. It refers to the acute angle that the terminal side of an angle tt makes with the x-axis. To find this number, we often reduce the given angle to an equivalent one between 0 and 2\u03c02\\pi by adding or subtracting multiples of 2\u03c02\\pi. Once we have the coterminal angle, the reference angle is based on which quadrant it lies in:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Quadrant I: reference angle = tt<\/li>\n\n\n\n<li>Quadrant II: reference angle = \u03c0\u2212t\\pi &#8211; t<\/li>\n\n\n\n<li>Quadrant III: reference angle = t\u2212\u03c0t &#8211; \\pi<\/li>\n\n\n\n<li>Quadrant IV: reference angle = 2\u03c0\u2212t2\\pi &#8211; t<\/li>\n<\/ul>\n\n\n\n<p>Negative angles rotate clockwise, so we add 2\u03c02\\pi repeatedly to get a positive coterminal angle. Non-radian numbers like 6 or -7 are interpreted as radians unless otherwise stated.<\/p>\n\n\n\n<p>By using these techniques, we determined reference angles and the terminal point on the unit circle accurately, reinforcing the importance of unit circle concepts in trigonometry.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner5-104.jpeg\" alt=\"\" class=\"wp-image-221015\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the terminal point P(x, y) on the unit circle determined by the given value of t. t = 3 pi\/2 Find the reference number for each value of t. The Correct Answer and Explanation is: 1. Terminal Point on the Unit Circle Given:t=3\u03c02t = \\frac{3\\pi}{2} On the unit circle, the coordinates of a point [&hellip;]<\/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-221013","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221013","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=221013"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221013\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221013"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221013"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}