![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-371.png\")
<\/figure>\n\n\n\nThe Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve for the exact value of sin\u2061(4\u03c03)\\sin\\left(\\frac{4\\pi}{3}\\right)sin(34\u03c0\u200b), we will follow a step-by-step trigonometric approach and explain each part clearly.<\/p>\n\n\n\n The angle given is 4\u03c03\\frac{4\\pi}{3}34\u03c0\u200b. This is a radian measure. To understand where it lies on the unit circle, we convert it to degrees:4\u03c03\u00d7180\u2218\u03c0=240\u2218\\frac{4\\pi}{3} \\times \\frac{180^\\circ}{\\pi} = 240^\\circ34\u03c0\u200b\u00d7\u03c0180\u2218\u200b=240\u2218<\/p>\n\n\n\n So, 4\u03c03\\frac{4\\pi}{3}34\u03c0\u200b radians is equivalent to 240\u2218240^\\circ240\u2218, which lies in the third quadrant<\/strong>.<\/p>\n\n\n\n In the third quadrant, the reference angle is found by subtracting 180\u2218180^\\circ180\u2218 from the angle:240\u2218\u2212180\u2218=60\u2218240^\\circ – 180^\\circ = 60^\\circ240\u2218\u2212180\u2218=60\u2218<\/p>\n\n\n\n This means that the reference angle is 60\u221860^\\circ60\u2218 (or \u03c03\\frac{\\pi}{3}3\u03c0\u200b radians).<\/p>\n\n\n\n In the third quadrant, the sine function is negative<\/strong> because in the unit circle, y-values (which represent sine) are negative below the x-axis.<\/p>\n\n\n\n From trigonometric values of special angles:sin\u2061(60\u2218)=sin\u2061(\u03c03)=32\\sin\\left(60^\\circ\\right) = \\sin\\left(\\frac{\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}sin(60\u2218)=sin(3\u03c0\u200b)=23\u200b\u200b<\/p>\n\n\n\n Since our angle is in the third quadrant where sine is negative:sin\u2061(4\u03c03)=\u221232\\sin\\left(\\frac{4\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}sin(34\u03c0\u200b)=\u221223\u200b\u200b<\/p>\n\n\n\n sin\u2061(4\u03c03)=\u221232\\boxed{\\sin\\left(\\frac{4\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}}sin(34\u03c0\u200b)=\u221223\u200b\u200b\u200b<\/p>\n\n\n\n To find the exact value of a trigonometric function for an angle in radians, we identify the quadrant in which the angle lies, determine the reference angle, use the known trigonometric values of that reference angle, and apply the correct sign based on the quadrant. In this case, since 4\u03c03\\frac{4\\pi}{3}34\u03c0\u200b is in the third quadrant and the sine of \u03c03\\frac{\\pi}{3}3\u03c0\u200b is 32\\frac{\\sqrt{3}}{2}23\u200b\u200b, we assign a negative sign to get the final answer \u221232-\\frac{\\sqrt{3}}{2}\u221223\u200b\u200b.<\/p>\n\n\n\n Find the exact value of . The Correct Answer and Explanation is: To solve for the exact value of sin\u2061(4\u03c03)\\sin\\left(\\frac{4\\pi}{3}\\right)sin(34\u03c0\u200b), we will follow a step-by-step trigonometric approach and explain each part clearly. Step 1: Understand the Angle The angle given is 4\u03c03\\frac{4\\pi}{3}34\u03c0\u200b. This is a radian measure. To understand where it lies on the unit […]<\/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-231759","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231759","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=231759"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231759\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231759"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231759"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231759"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Understand the Angle<\/strong><\/h3>\n\n\n\n
\n\n\n\nStep 2: Reference Angle<\/strong><\/h3>\n\n\n\n
\n\n\n\nStep 3: Sign of the Sine Function<\/strong><\/h3>\n\n\n\n
\n\n\n\nStep 4: Exact Value of sin\u2061(60\u2218)\\sin(60^\\circ)sin(60\u2218)<\/strong><\/h3>\n\n\n\n
\n\n\n\nFinal Answer:<\/strong><\/h3>\n\n\n\n
\n\n\n\nConclusion:<\/strong><\/h3>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-965.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"