{"id":217614,"date":"2025-05-22T08:11:58","date_gmt":"2025-05-22T08:11:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217614"},"modified":"2025-05-22T08:12:00","modified_gmt":"2025-05-22T08:12:00","slug":"find-the-exact-value-of-the-trigonometric-function-at-the-given-real-number","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/find-the-exact-value-of-the-trigonometric-function-at-the-given-real-number\/","title":{"rendered":"Find the exact value of the trigonometric function at the given real number."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Find the exact value of the trigonometric function at the given real number.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(a) sin(5pi\/3) <strong>__<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) cos(23pi\/3) <strong>__<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(c) tan(5pi\/3) <strong><em>_<\/em><\/strong><\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>Determine the coefficient of<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">(a) x 2y 3 z 2 in (x + y + z) 7<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(b) x 2y 3 z 2 in (x + 2y + z)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s solve the problems one by one with clear steps and explanations.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Trigonometric Function Values<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">To solve these, we reduce the angles to an equivalent angle between $0$ and $2\\pi$ and then use the unit circle.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(a) $\\sin\\left(\\frac{5\\pi}{3}\\right)$<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: $\\frac{5\\pi}{3}$ is in radians.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since $2\\pi = \\frac{6\\pi}{3}$, we see:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{5\\pi}{3} &lt; 2\\pi \\Rightarrow \\text{Angle lies within one full circle}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Find reference angle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since $\\frac{5\\pi}{3}$ is in the <strong>4th quadrant<\/strong>, the reference angle is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>2\\pi &#8211; \\frac{5\\pi}{3} = \\frac{\\pi}{3}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 3: Use unit circle for $\\sin(\\pi\/3) = \\frac{\\sqrt{3}}{2}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since sine is <strong>negative<\/strong> in the 4th quadrant:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\boxed{\\sin\\left(\\frac{5\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(b) $\\cos\\left(\\frac{23\\pi}{3}\\right)$<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Step 1: Simplify the angle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{23\\pi}{3} \\div 2\\pi = \\frac{23\\pi}{3} \\cdot \\frac{1}{2\\pi} = \\frac{23}{6} \\Rightarrow \\text{That&#8217;s 3 full circles and } \\frac{5\\pi}{3}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{23\\pi}{3} \\equiv \\frac{5\\pi}{3} \\mod 2\\pi<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Step 2: Use cosine of $\\frac{5\\pi}{3}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As we saw earlier, reference angle is $\\frac{\\pi}{3}$, cosine is <strong>positive<\/strong> in 4th quadrant:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\cos\\left(\\frac{5\\pi}{3}\\right) = \\cos\\left(\\frac{\\pi}{3}\\right) = \\frac{1}{2}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\boxed{\\cos\\left(\\frac{23\\pi}{3}\\right) = \\frac{1}{2}}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(c) $\\tan\\left(\\frac{5\\pi}{3}\\right)$<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Use:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\tan\\left(\\frac{5\\pi}{3}\\right) = \\frac{\\sin\\left(\\frac{5\\pi}{3}\\right)}{\\cos\\left(\\frac{5\\pi}{3}\\right)} = \\frac{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} = -\\sqrt{3}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\boxed{\\tan\\left(\\frac{5\\pi}{3}\\right) = -\\sqrt{3}}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Coefficient in Multinomial Expansion<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We apply the <strong>Multinomial Theorem<\/strong> for expressions of the form $(x + y + z)^n$:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>(x + y + z)^n = \\sum \\frac{n!}{a!b!c!} x^a y^b z^c<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where $a + b + c = n$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(a) Coefficient of $x^2 y^3 z^2$ in $(x + y + z)^7$<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">We use:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$a = 2, b = 3, c = 2$<\/li>\n\n\n\n<li>$a + b + c = 7$, so valid.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\text{Coefficient} = \\frac{7!}{2!3!2!} = \\frac{5040}{2 \\cdot 6 \\cdot 2} = \\frac{5040}{24} = \\boxed{210}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>(b) Coefficient of $x^2 y^3 z^2$ in $(x + 2y + z)^7$<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">This is not just a multinomial coefficient. Since there is a coefficient inside the expression (i.e., 2 with $y$), we compute:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Use multinomial coefficient:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{7!}{2!3!2!} = 210 \\quad \\text{(same as above)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But account for the extra factor from $(2y)^3 = 8y^3$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So the actual coefficient is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>210 \\cdot 8 = \\boxed{1680}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Detailed Explanation (300+ Words)<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In part one, we deal with trigonometric functions of angles given in radians. Since trigonometric functions are periodic, we reduce angles to a principal value within $0$ to $2\\pi$. The sine, cosine, and tangent functions follow known rules in each quadrant of the unit circle. For example, $\\sin\\left(\\frac{5\\pi}{3}\\right)$ lies in the 4th quadrant where sine is negative. The reference angle is $\\frac{\\pi}{3}$, and the sine of that is $\\frac{\\sqrt{3}}{2}$, so the final value is negative. Similarly, cosine in the 4th quadrant is positive, and tangent is negative. When evaluating $\\cos\\left(\\frac{23\\pi}{3}\\right)$, we reduce the angle by subtracting multiples of $2\\pi$ until it falls within one circle, and we again land at $\\frac{5\\pi}{3}$, giving the same cosine value as before.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In part two, we use the <strong>multinomial theorem<\/strong>, which generalizes the binomial theorem to more than two terms. The multinomial coefficient $\\frac{n!}{a!b!c!}$ tells us how many times the term $x^a y^b z^c$ appears in the expansion of $(x + y + z)^n$. In problem (a), we compute the coefficient of $x^2 y^3 z^2$ in the expansion of $(x + y + z)^7$, resulting in $\\frac{7!}{2!3!2!} = 210$. In problem (b), because one term has a coefficient (i.e., $2y$), we must include this extra multiplier. So, $(2y)^3 = 8y^3$, meaning the coefficient becomes $210 \\cdot 8 = 1680$. This illustrates the importance of paying attention to both exponents and numerical coefficients in algebraic expansions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-19.png\" alt=\"\" class=\"wp-image-217615\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the exact value of the trigonometric function at the given real number. (a) sin(5pi\/3) __ (b) cos(23pi\/3) __ (c) tan(5pi\/3) _ (a) x 2y 3 z 2 in (x + y + z) 7 (b) x 2y 3 z 2 in (x + 2y + z) The Correct Answer and Explanation is: Let&#8217;s solve [&hellip;]<\/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-217614","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217614","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=217614"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217614\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217614"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217614"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217614"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}