{"id":30879,"date":"2025-06-21T20:30:34","date_gmt":"2025-06-21T20:30:34","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=30879"},"modified":"2025-06-21T20:30:35","modified_gmt":"2025-06-21T20:30:35","slug":"use-the-unit-circle-to-evaluate-the-function","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/use-the-unit-circle-to-evaluate-the-function\/","title":{"rendered":"Use the unit circle to evaluate the function."},"content":{"rendered":"\n<p>Use the unit circle to evaluate the function. Tan 7pie\/6<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate <strong>tan(7\u03c0\/6)<\/strong> using the unit circle:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Correct Answer:<\/strong><\/h3>\n\n\n\n<p>\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{1}{\\sqrt{3}} \\cdot (-1)^2 = \\frac{1}{\\sqrt{3}} \\cdot 1 = \\frac{1}{\\sqrt{3}} \\text{ with the sign } (-), \\text{ so } \\boxed{\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{1}{\\sqrt{3}} \\cdot (-1) = \\boxed{\\frac{1}{\\sqrt{3}}(-1) = \\boxed{\\frac{1}{\\sqrt{3}} = -\\frac{\\sqrt{3}}{3}}}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83e\udde0 <strong>Step-by-Step Explanation:<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Understand the angle:<\/strong> 7\u03c06\u00a0is\u00a0in\u00a0radians.\u00a0To\u00a0understand\u00a0its\u00a0location\u00a0on\u00a0the\u00a0unit\u00a0circle,\u00a0convert\u00a0it\u00a0to\u00a0degrees:\\frac{7\\pi}{6} \\text{ is in radians. To understand its location on the unit circle, convert it to degrees:}67\u03c0\u200b\u00a0is\u00a0in\u00a0radians.\u00a0To\u00a0understand\u00a0its\u00a0location\u00a0on\u00a0the\u00a0unit\u00a0circle,\u00a0convert\u00a0it\u00a0to\u00a0degrees: 7\u03c06\u22c5180\u2218\u03c0=210\u2218\\frac{7\\pi}{6} \\cdot \\frac{180^\\circ}{\\pi} = 210^\\circ67\u03c0\u200b\u22c5\u03c0180\u2218\u200b=210\u2218<\/li>\n\n\n\n<li><strong>Find the reference angle:<\/strong><br>The reference angle is the acute angle formed with the x-axis. Since 210\u00b0 is in the third quadrant: 210\u2218\u2212180\u2218=30\u2218210^\\circ &#8211; 180^\\circ = 30^\\circ210\u2218\u2212180\u2218=30\u2218 So the reference angle is 30\u00b0 (or \u03c0\/6 radians).<\/li>\n\n\n\n<li><strong>Use known values from the unit circle:<\/strong><br>For the reference angle \u03c0\/6:\n<ul class=\"wp-block-list\">\n<li>sin\u2061(\u03c0\/6)=12\\sin(\\pi\/6) = \\frac{1}{2}sin(\u03c0\/6)=21\u200b<\/li>\n\n\n\n<li>cos\u2061(\u03c0\/6)=32\\cos(\\pi\/6) = \\frac{\\sqrt{3}}{2}cos(\u03c0\/6)=23\u200b\u200b<\/li>\n\n\n\n<li>tan\u2061(\u03c0\/6)=13=33\\tan(\\pi\/6) = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}tan(\u03c0\/6)=3\u200b1\u200b=33\u200b\u200b<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Determine the sign of tangent in the third quadrant:<\/strong><br>In the third quadrant, both sine and cosine are negative, and since tangent is sine divided by cosine, the negatives cancel out: tan\u2061(7\u03c06)=\u221212\u221232=13=33\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}tan(67\u03c0\u200b)=\u221223\u200b\u200b\u221221\u200b\u200b=3\u200b1\u200b=33\u200b\u200b<\/li>\n\n\n\n<li><strong>Final step \u2013 correct sign:<\/strong><br>Since both sine and cosine are negative in quadrant III, their ratio is <strong>positive<\/strong>. So: tan\u2061(7\u03c06)=33\\boxed{\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{\\sqrt{3}}{3}}tan(67\u03c0\u200b)=33\u200b\u200b\u200b<\/li>\n<\/ol>\n\n\n\n<p>However, there is a sign mistake in the earlier step. Let&#8217;s correct:<\/p>\n\n\n\n<p>At 210\u00b0, <strong>sine and cosine are both negative<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin\u2061(210\u2218)=\u221212\\sin(210^\\circ) = -\\frac{1}{2}sin(210\u2218)=\u221221\u200b<\/li>\n\n\n\n<li>cos\u2061(210\u2218)=\u221232\\cos(210^\\circ) = -\\frac{\\sqrt{3}}{2}cos(210\u2218)=\u221223\u200b\u200b<\/li>\n<\/ul>\n\n\n\n<p>So:tan\u2061(210\u2218)=\u221212\u221232=13=33\\tan(210^\\circ) = \\frac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}} = \\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}tan(210\u2218)=\u221223\u200b\u200b\u221221\u200b\u200b=3\u200b1\u200b=33\u200b\u200b<\/p>\n\n\n\n<p>Therefore, the final correct answer is:tan\u2061(7\u03c06)=33\\boxed{\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{\\sqrt{3}}{3}}tan(67\u03c0\u200b)=33\u200b\u200b\u200b<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-218.jpeg\" alt=\"\" class=\"wp-image-30880\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-218.jpeg 722w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-218-212x300.jpeg 212w\" sizes=\"auto, (max-width: 722px) 100vw, 722px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Use the unit circle to evaluate the function. Tan 7pie\/6 The Correct Answer and Explanation is: To evaluate tan(7\u03c0\/6) using the unit circle: \u2705 Correct Answer: \\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{1}{\\sqrt{3}} \\cdot (-1)^2 = \\frac{1}{\\sqrt{3}} \\cdot 1 = \\frac{1}{\\sqrt{3}} \\text{ with the sign } (-), \\text{ so } \\boxed{\\tan\\left(\\frac{7\\pi}{6}\\right) = \\frac{1}{\\sqrt{3}} \\cdot (-1) = \\boxed{\\frac{1}{\\sqrt{3}}(-1) = \\boxed{\\frac{1}{\\sqrt{3}} [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-30879","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30879","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=30879"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30879\/revisions"}],"predecessor-version":[{"id":30881,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30879\/revisions\/30881"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=30879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=30879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=30879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}