Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Appendix D - Trigonometry - D Exercises - Page A32: 24


$sin\frac{4\pi}{3}=-\frac{\sqrt 3}{2}$,$csc\frac{4\pi}{3}=-\frac{2\sqrt 3}{3}$,$cos\frac{4\pi}{3}=-\frac{1}{2}$, $sec\frac{4\pi}{3}=-2$,$tan\frac{4\pi}{3}=\sqrt 3$, $cot\frac{4\pi}{3}=\frac{\sqrt 3}{3}$

Work Step by Step

Since, $\frac{4\pi}{3}$ lies on third quadrant, thus its reference angle will be $\frac{4\pi}{3}-\pi-=\frac{\pi}{3}$, which can be considered as common angle to all trigonometric ratios. The trigonometric ratios and their inverse trigonometric ratios are given as follows: $sin\frac{4\pi}{3}=-sin\frac{\pi}{3}=-\frac{\sqrt 3}{2}$ $csc\frac{4\pi}{3}=\frac{1}{sin\frac{4\pi}{3}}=-\frac{2\sqrt 3}{3}$ $cos\frac{4\pi}{3}=-cos\frac{\pi}{3}=-\frac{1}{2}$ $sec\frac{4\pi}{3}=\frac{1}{cos\frac{4\pi}{3}}=-2$ $tan\frac{4\pi}{3}=tan\frac{\pi}{3}=\sqrt 3$ $cot\frac{4\pi}{3}=\frac{1}{tan\frac{4\pi}{3}}=\frac{\sqrt 3}{3}$
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