{"id":41016,"date":"2025-06-28T03:49:15","date_gmt":"2025-06-28T03:49:15","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=41016"},"modified":"2025-06-28T03:49:16","modified_gmt":"2025-06-28T03:49:16","slug":"find-the-exact-value-of-the-trigonometric-expression-without-the-use-of-a-calculator","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-value-of-the-trigonometric-expression-without-the-use-of-a-calculator\/","title":{"rendered":"Find the exact value of the trigonometric expression without the use of a calculator."},"content":{"rendered":"\n<p>Find the exact value of the trigonometric expression without the use of a calculator. sin(pi\/12) cos(7pi\/12) + cos(pi\/12) sin(7pi\/12) =<\/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<h3 class=\"wp-block-heading\">Correct Answer:<\/h3>\n\n\n\n<p>We are asked to find the exact value of:sin\u2061(\u03c012)cos\u2061(7\u03c012)+cos\u2061(\u03c012)sin\u2061(7\u03c012)\\sin\\left(\\frac{\\pi}{12}\\right)\\cos\\left(\\frac{7\\pi}{12}\\right) + \\cos\\left(\\frac{\\pi}{12}\\right)\\sin\\left(\\frac{7\\pi}{12}\\right)sin(12\u03c0\u200b)cos(127\u03c0\u200b)+cos(12\u03c0\u200b)sin(127\u03c0\u200b)<\/p>\n\n\n\n<p>This expression matches the identity for the sine of a sum of angles:sin\u2061(A+B)=sin\u2061(A)cos\u2061(B)+cos\u2061(A)sin\u2061(B)\\sin(A + B) = \\sin(A)\\cos(B) + \\cos(A)\\sin(B)sin(A+B)=sin(A)cos(B)+cos(A)sin(B)<\/p>\n\n\n\n<p>In this case:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A=\u03c012A = \\frac{\\pi}{12}A=12\u03c0\u200b<\/li>\n\n\n\n<li>B=7\u03c012B = \\frac{7\\pi}{12}B=127\u03c0\u200b<\/li>\n<\/ul>\n\n\n\n<p>Thus, applying the identity:sin\u2061(\u03c012+7\u03c012)=sin\u2061(8\u03c012)=sin\u2061(2\u03c03)\\sin\\left(\\frac{\\pi}{12} + \\frac{7\\pi}{12}\\right) = \\sin\\left(\\frac{8\\pi}{12}\\right) = \\sin\\left(\\frac{2\\pi}{3}\\right)sin(12\u03c0\u200b+127\u03c0\u200b)=sin(128\u03c0\u200b)=sin(32\u03c0\u200b)<\/p>\n\n\n\n<p>We know that:sin\u2061(2\u03c03)=sin\u2061(\u03c0\u2212\u03c03)\\sin\\left(\\frac{2\\pi}{3}\\right) = \\sin\\left(\\pi &#8211; \\frac{\\pi}{3}\\right)sin(32\u03c0\u200b)=sin(\u03c0\u22123\u03c0\u200b)<\/p>\n\n\n\n<p>Using the identity for sine of supplementary angles:sin\u2061(\u03c0\u2212x)=sin\u2061(x)\\sin(\\pi &#8211; x) = \\sin(x)sin(\u03c0\u2212x)=sin(x)<\/p>\n\n\n\n<p>Therefore:sin\u2061(2\u03c03)=sin\u2061(\u03c03)=32\\sin\\left(\\frac{2\\pi}{3}\\right) = \\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}sin(32\u03c0\u200b)=sin(3\u03c0\u200b)=23\u200b\u200b<\/p>\n\n\n\n<p><strong>Final exact value<\/strong>:32\\boxed{\\frac{\\sqrt{3}}{2}}23\u200b\u200b\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Detailed Explanation:<\/h3>\n\n\n\n<p>This problem tests knowledge of trigonometric identities and simplifications involving radians. The given expression fits the standard identity for the sine of a sum of angles, which simplifies the calculation significantly.<\/p>\n\n\n\n<p>First, identify the structure:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The expression has sin\u2061\\sinsin of one angle multiplied by cos\u2061\\coscos of another and vice versa, which is the exact form of sin\u2061(A+B)\\sin(A + B)sin(A+B).<\/li>\n\n\n\n<li>Plug in the angles: A=\u03c012A = \\frac{\\pi}{12}A=12\u03c0\u200b and B=7\u03c012B = \\frac{7\\pi}{12}B=127\u03c0\u200b.<\/li>\n\n\n\n<li>Their sum is 8\u03c012=2\u03c03\\frac{8\\pi}{12} = \\frac{2\\pi}{3}128\u03c0\u200b=32\u03c0\u200b.<\/li>\n<\/ul>\n\n\n\n<p>To find sin\u2061(2\u03c03)\\sin\\left(\\frac{2\\pi}{3}\\right)sin(32\u03c0\u200b), note that 2\u03c03\\frac{2\\pi}{3}32\u03c0\u200b lies in the second quadrant where sine is positive. Using reference angles, sin\u2061(2\u03c03)\\sin(\\frac{2\\pi}{3})sin(32\u03c0\u200b) simplifies to sin\u2061(\u03c03)\\sin(\\frac{\\pi}{3})sin(3\u03c0\u200b).<\/p>\n\n\n\n<p>The known exact value for sin\u2061(\u03c03)\\sin(\\frac{\\pi}{3})sin(3\u03c0\u200b) is 32\\frac{\\sqrt{3}}{2}23\u200b\u200b.<\/p>\n\n\n\n<p>Thus, without needing a calculator, the expression simplifies directly to 32\\frac{\\sqrt{3}}{2}23\u200b\u200b.<\/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-banner8-1301.jpeg\" alt=\"\" class=\"wp-image-41017\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1301.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1301-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1301-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of the trigonometric expression without the use of a calculator. sin(pi\/12) cos(7pi\/12) + cos(pi\/12) sin(7pi\/12) = The Correct Answer and Explanation is: Correct Answer: We are asked to find the exact value of:sin\u2061(\u03c012)cos\u2061(7\u03c012)+cos\u2061(\u03c012)sin\u2061(7\u03c012)\\sin\\left(\\frac{\\pi}{12}\\right)\\cos\\left(\\frac{7\\pi}{12}\\right) + \\cos\\left(\\frac{\\pi}{12}\\right)\\sin\\left(\\frac{7\\pi}{12}\\right)sin(12\u03c0\u200b)cos(127\u03c0\u200b)+cos(12\u03c0\u200b)sin(127\u03c0\u200b) This expression matches the identity for the sine of a sum of angles:sin\u2061(A+B)=sin\u2061(A)cos\u2061(B)+cos\u2061(A)sin\u2061(B)\\sin(A + B) = [&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-41016","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41016","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=41016"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41016\/revisions"}],"predecessor-version":[{"id":41018,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41016\/revisions\/41018"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=41016"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=41016"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=41016"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}