Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.6 - Inverse Trigonometric Functions - Exercises 7.6 - Page 420: 6

Answer

$a.\displaystyle \quad \frac{\pi}{4}$ $b.\displaystyle \quad -\frac{\pi}{3}$ $c.\displaystyle \quad \frac{\pi}{6}$

Work Step by Step

$y=\csc^{-1}x$ is the number in $[-\pi/2,0)\ \cup\ (0, \pi/2]$ for which $\csc y=x.$ $(\displaystyle \csc y=x \Leftrightarrow \sin y=\frac{1}{x})$ In quadrant I, we have $(\mathrm{a})$ $\displaystyle \sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\quad\Rightarrow\quad\csc\frac{\pi}{4}=\sqrt{2}\qquad \Rightarrow\quad\csc^{-1}\sqrt{2}=\frac{\pi}{4}$ $(b)$ $\displaystyle \sin\frac{\pi}{3}=\frac{ \sqrt{3}}{2}\quad\Rightarrow\quad \sin(-\frac{\pi}{3})=-\frac{ \sqrt{3}}{2} \quad$ $\displaystyle \Rightarrow\quad \csc(-\frac{\pi}{3})=-\frac{2}{\sqrt{3}}\quad\Rightarrow\quad\csc^{-1}(-\frac{2}{\sqrt{3}})=-\frac{\pi}{3}$ $(c)$ $\displaystyle \sin\frac{\pi}{6}=\frac{1}{2}\quad\Rightarrow\quad \csc(\frac{\pi}{6})=\frac{2}{\sqrt{3}}\quad\Rightarrow\quad\csc^{-1}(\frac{2}{1})=\frac{\pi}{6}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.