{"id":41077,"date":"2025-06-28T07:09:28","date_gmt":"2025-06-28T07:09:28","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=41077"},"modified":"2025-06-28T07:09:29","modified_gmt":"2025-06-28T07:09:29","slug":"find-the-exact-value-for-each-trigonometric-function","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-value-for-each-trigonometric-function\/","title":{"rendered":"Find the exact value for each trigonometric function"},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">Find the exact value for each trigonometric function<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"738\" height=\"422\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-435.png\" alt=\"\" class=\"wp-image-41078\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-435.png 738w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-435-300x172.png 300w\" sizes=\"auto, (max-width: 738px) 100vw, 738px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a) sin\u2061(2\u03c03)=32\\sin\\left(\\frac{2\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}<\/li>\n\n\n\n<li>(b) sin\u2061(4\u03c03)=\u221232\\sin\\left(\\frac{4\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}<\/li>\n\n\n\n<li>(c) sin\u2061(9\u03c02)=1\\sin\\left(\\frac{9\\pi}{2}\\right) = 1<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>To solve these problems, we use the unit circle and the concept of reference angles. The sine function corresponds to the y-coordinate of a point on the unit circle.<\/p>\n\n\n\n<p><strong>(a) <\/strong>sin\u2061(2\u03c03)\\sin\\left(\\frac{2\\pi}{3}\\right) The angle 2\u03c03\\frac{2\\pi}{3} lies in the second quadrant. The reference angle is \u03c0\u22122\u03c03=\u03c03\\pi &#8211; \\frac{2\\pi}{3} = \\frac{\\pi}{3}. In the second quadrant, sine values are positive. Since sin\u2061(\u03c03)=32\\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}, we have:<\/p>\n\n\n\n<p>sin\u2061(2\u03c03)=32\\sin\\left(\\frac{2\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}<\/p>\n\n\n\n<p><strong>(b) <\/strong>sin\u2061(4\u03c03)\\sin\\left(\\frac{4\\pi}{3}\\right) The angle 4\u03c03\\frac{4\\pi}{3} lies in the third quadrant. The reference angle is 4\u03c03\u2212\u03c0=\u03c03\\frac{4\\pi}{3} &#8211; \\pi = \\frac{\\pi}{3}. In the third quadrant, sine values are negative. So,<\/p>\n\n\n\n<p>sin\u2061(4\u03c03)=\u221232\\sin\\left(\\frac{4\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}<\/p>\n\n\n\n<p><strong>(c) <\/strong>sin\u2061(9\u03c02)\\sin\\left(\\frac{9\\pi}{2}\\right) This angle is larger than 2\u03c02\\pi, so we reduce it by subtracting full rotations. Each full rotation is 2\u03c02\\pi, or 4\u03c02\\frac{4\\pi}{2}.<\/p>\n\n\n\n<p>9\u03c02\u22124\u03c0=9\u03c02\u22128\u03c02=\u03c02\\frac{9\\pi}{2} &#8211; 4\\pi = \\frac{9\\pi}{2} &#8211; \\frac{8\\pi}{2} = \\frac{\\pi}{2}<\/p>\n\n\n\n<p>Now we evaluate sin\u2061(\u03c02)\\sin\\left(\\frac{\\pi}{2}\\right), which equals 1. So,<\/p>\n\n\n\n<p>sin\u2061(9\u03c02)=1\\sin\\left(\\frac{9\\pi}{2}\\right) = 1<\/p>\n\n\n\n<p>These problems test your understanding of angle placement on the unit circle, quadrant rules, and reference angles. Mastery of these ideas is essential for solving trigonometric equations and analyzing periodic behavior in advanced mathematics.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1318.jpeg\" alt=\"\" class=\"wp-image-41079\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1318.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1318-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1318-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value for each trigonometric function Final Answers: Explanation To solve these problems, we use the unit circle and the concept of reference angles. The sine function corresponds to the y-coordinate of a point on the unit circle. (a) sin\u2061(2\u03c03)\\sin\\left(\\frac{2\\pi}{3}\\right) The angle 2\u03c03\\frac{2\\pi}{3} lies in the second quadrant. The reference angle is \u03c0\u22122\u03c03=\u03c03\\pi [&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-41077","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41077","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=41077"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41077\/revisions"}],"predecessor-version":[{"id":41115,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41077\/revisions\/41115"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=41077"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=41077"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=41077"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}