Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 6 - Inverse Circular Functions and Trigonometric Equations - Section 6.1 Inverse Circular Functions - 6.1 Exercises - Page 264: 18


The value of $y$ in this case is $$y=-\frac{\pi}{4}$$

Work Step by Step

$$y=\tan^{-1} (-1)$$ First, we see that the domain of inverse tangent function is $[-\infty,\infty]$. Therefore, in fact when we deal with inverse tangent function, we do not need to do this checking step. The range of inverse tangent function is $(-\frac{\pi}{2},\frac{\pi}{2})$. In other words, $y\in(-\frac{\pi}{2},\frac{\pi}{2})$. We can rewrite $y=\tan^{-1}(-1)$ into $\tan y=-1$ We know that $$\tan\frac{\pi}{4}=1$$ which means $$-\tan\frac{\pi}{4}=-1$$ $$\tan(-\frac{\pi}{4})=-1$$ Also, we see that $-\frac{\pi}{4}\in(-\frac{\pi}{2},\frac{\pi}{2})$ Therefore, the exact value of $y$ here is $$y=-\frac{\pi}{4}$$
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