Trigonometry 7th Edition

Published by Cengage Learning
ISBN 10: 1111826854
ISBN 13: 978-1-11182-685-7

Chapter 3 - Section 3.2 - Radians and Degrees - 3.2 Problem Set - Page 133: 61



Work Step by Step

$\dfrac{2\pi}{3}$ is an angle whose terminal side is in Quadrant II. The reference angle of an angle $\theta$ in Quadrant II can be found using the formula $\pi - \theta$. Thus, the reference angle of the given angle is: $=\pi - \dfrac{2\pi}{3} \\=\dfrac{3\pi}{3} -\dfrac{2\pi}{3} \\=\dfrac{\pi}{3}$ With a reference angle of $\dfrac{\pi}{3}$, then the cosine value of $\dfrac{2\pi}{3}$ will be the same as $\dfrac{\pi}{3}$ except possibly for its sign. The value of cosine of $\dfrac{\pi}{3}$ is $\dfrac{1}{2}$. However, since $\dfrac{2\pi}{3}$ is in Quadrant II, its cosiine is negative. Thus, $\cos{(\frac{2\pi}{3})} = -\dfrac{1}{2}$. RECALL: $\sec{\theta} = \dfrac{1}{\cos{\theta}}$ Using the formula above gives: $\sec{(\frac{2\pi}{3})}=\dfrac{1}{\cos{(\frac{2\pi}{3})}}=\dfrac{1}{-\frac{1}{2}}=1 \cdot \dfrac{-2}{1} = -2$
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