Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 6 - Trigonometric Functions - 6.3 Properties of the Trigonometric Functions - 6.3 Assess Your Understanding - Page 395: 76



Work Step by Step

We know that $\sin$, $\csc$, $\tan$ are odd functions, which means $f(-\theta)=-f(\theta).$ We also know that $\cos$, $\sec$, are even functions, which means $f(-\theta)=f(\theta).$ Therefore, $\csc{\left(-\frac{\pi}{3}\right)}\\ =-\csc{\left(\frac{\pi}{3}\right)}\\ =-\dfrac{1}{\sin(\frac{\pi}{3})}\\ =-\dfrac{1}{\frac{\sqrt3}{2}}\\ =-\frac{2}{\sqrt3}\\ =-\frac{2}{\sqrt3} \cdot \frac{\sqrt3}{\sqrt3}\\ =-\frac{2\sqrt3}{3}$
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