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: 73



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, $\csc{\left(-\frac{\pi}{4}\right)}\\ =-\csc{\left(\frac{\pi}{4}\right)}\\ =-\dfrac{1}{\sin(\frac{\pi}{4})}\\ =-\dfrac{1}{\frac{\sqrt2}{2}}\\ =-\dfrac{2}{\sqrt2}\\ =-\dfrac{2}{\sqrt2} \cdot \dfrac{\sqrt2}{\sqrt2}\\ =-\dfrac{2\sqrt2}{2}\\ =-\sqrt2$
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