{"id":250245,"date":"2025-07-10T08:28:16","date_gmt":"2025-07-10T08:28:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=250245"},"modified":"2025-07-10T08:28:18","modified_gmt":"2025-07-10T08:28:18","slug":"if-t-7%cf%80-6-find-the-point-xy-on-the-unit-circle-that-corresponds-to-the-real-number-t","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/if-t-7%cf%80-6-find-the-point-xy-on-the-unit-circle-that-corresponds-to-the-real-number-t\/","title":{"rendered":"If t= 7\u03c0\/6 find the point (x,y) on the unit circle that corresponds to the real number t."},"content":{"rendered":"\n

If t= 7\u03c0\/6 find the point (x,y) on the unit circle that corresponds to the real number t. Calculate sin t, cos t, and tan t. Answer in the exact form<\/p>\n\n\n\n

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

To find the point (x,y)(x, y)(x,y) on the unit circle corresponding to t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b, we will first determine the reference angle, and then use the trigonometric functions to find sin\u2061t\\sin tsint, cos\u2061t\\cos tcost, and tan\u2061t\\tan ttant.<\/p>\n\n\n\n

Step 1: Reference Angle<\/h3>\n\n\n\n

The angle t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b is in the third quadrant of the unit circle because it is between \u03c0\\pi\u03c0 and 3\u03c0\/23\\pi\/23\u03c0\/2. To find the reference angle, we subtract \u03c0\\pi\u03c0 from ttt:Reference angle=7\u03c06\u2212\u03c0=7\u03c06\u22126\u03c06=\u03c06\\text{Reference angle} = \\frac{7\\pi}{6} – \\pi = \\frac{7\\pi}{6} – \\frac{6\\pi}{6} = \\frac{\\pi}{6}Reference angle=67\u03c0\u200b\u2212\u03c0=67\u03c0\u200b\u221266\u03c0\u200b=6\u03c0\u200b<\/p>\n\n\n\n

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

Step 2: Coordinates on the Unit Circle<\/h3>\n\n\n\n

On the unit circle, the point corresponding to the angle t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b will have coordinates (x,y)(x, y)(x,y) where:<\/p>\n\n\n\n

    \n
  • x=cos\u2061tx = \\cos tx=cost<\/li>\n\n\n\n
  • y=sin\u2061ty = \\sin ty=sint<\/li>\n<\/ul>\n\n\n\n

    The coordinates for the reference angle \u03c06\\frac{\\pi}{6}6\u03c0\u200b are:(cos\u2061\u03c06,sin\u2061\u03c06)=(32,12)\\left( \\cos \\frac{\\pi}{6}, \\sin \\frac{\\pi}{6} \\right) = \\left( \\frac{\\sqrt{3}}{2}, \\frac{1}{2} \\right)(cos6\u03c0\u200b,sin6\u03c0\u200b)=(23\u200b\u200b,21\u200b)<\/p>\n\n\n\n

    Since the angle is in the third quadrant, both cos\u2061t\\cos tcost and sin\u2061t\\sin tsint are negative:x=\u221232,y=\u221212x = -\\frac{\\sqrt{3}}{2}, \\quad y = -\\frac{1}{2}x=\u221223\u200b\u200b,y=\u221221\u200b<\/p>\n\n\n\n

    Thus, the point on the unit circle corresponding to t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b is:(x,y)=(\u221232,\u221212)(x, y) = \\left( -\\frac{\\sqrt{3}}{2}, -\\frac{1}{2} \\right)(x,y)=(\u221223\u200b\u200b,\u221221\u200b)<\/p>\n\n\n\n

    Step 3: Calculate Trigonometric Functions<\/h3>\n\n\n\n

    We now calculate sin\u2061t\\sin tsint, cos\u2061t\\cos tcost, and tan\u2061t\\tan ttant using the values from the unit circle.<\/p>\n\n\n\n

      \n
    1. Sine<\/strong>: The sine of t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b is the yyy-coordinate: sin\u2061t=\u221212\\sin t = -\\frac{1}{2}sint=\u221221\u200b<\/li>\n\n\n\n
    2. Cosine<\/strong>: The cosine of t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b is the xxx-coordinate: cos\u2061t=\u221232\\cos t = -\\frac{\\sqrt{3}}{2}cost=\u221223\u200b\u200b<\/li>\n\n\n\n
    3. Tangent<\/strong>: The tangent is the ratio of sine to cosine: tan\u2061t=sin\u2061tcos\u2061t=\u221212\u221232=13=33\\tan t = \\frac{\\sin t}{\\cos t} = \\frac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}tant=costsint\u200b=\u221223\u200b\u200b\u221221\u200b\u200b=3\u200b1\u200b=33\u200b\u200b<\/li>\n<\/ol>\n\n\n\n

      Final Answer<\/h3>\n\n\n\n
        \n
      • The point on the unit circle corresponding to t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b is (\u221232,\u221212)\\left( -\\frac{\\sqrt{3}}{2}, -\\frac{1}{2} \\right)(\u221223\u200b\u200b,\u221221\u200b).<\/li>\n\n\n\n
      • sin\u2061t=\u221212\\sin t = -\\frac{1}{2}sint=\u221221\u200b<\/li>\n\n\n\n
      • cos\u2061t=\u221232\\cos t = -\\frac{\\sqrt{3}}{2}cost=\u221223\u200b\u200b<\/li>\n\n\n\n
      • tan\u2061t=33\\tan t = \\frac{\\sqrt{3}}{3}tant=33\u200b\u200b<\/li>\n<\/ul>\n\n\n\n

        This result provides the exact values of the trigonometric functions based on the angle t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b, calculated using the unit circle.<\/p>\n\n\n\n

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

        If t= 7\u03c0\/6 find the point (x,y) on the unit circle that corresponds to the real number t. Calculate sin t, cos t, and tan t. Answer in the exact form The Correct Answer and Explanation is: To find the point (x,y)(x, y)(x,y) on the unit circle corresponding to t=7\u03c06t = \\frac{7\\pi}{6}t=67\u03c0\u200b, we will first […]<\/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-250245","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250245","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=250245"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250245\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=250245"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=250245"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=250245"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}