Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - 7.5 Inverse Circular Functions - 7.5 Exercises: 30

Answer

$y=\displaystyle \frac{\pi}{4}$

Work Step by Step

$y=\csc^{-1}x$ Domain: $(-\infty, -1]\cup[1, \infty)$ Range: $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ Quadrants (unit circle): I and IV --------- $y$ is the number from $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$ such that $\csc y=\sqrt{2}\quad (\displaystyle \sin y=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2})$ $\displaystyle \sin\frac{\pi}{4}=\frac{1}{\sqrt{2}} \Rightarrow \csc (\displaystyle \frac{\pi}{4})=\sqrt{2}$, and $\displaystyle \frac{\pi}{4}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$. So, $y=\displaystyle \frac{\pi}{4}$
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