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
$(\mathrm{a})$
$\displaystyle \sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\quad\Rightarrow\quad\csc\frac{\pi}{4}=\sqrt{2}\qquad \Rightarrow\quad\csc^{-1}\sqrt{2}=\frac{\pi}{4}$
$(b)$
$\displaystyle \sin\frac{\pi}{3}=\frac{ \sqrt{3}}{2}\quad\Rightarrow\quad \sin(-\frac{\pi}{3})=-\frac{ \sqrt{3}}{2} \quad$
$\displaystyle \Rightarrow\quad \csc(-\frac{\pi}{3})=-\frac{2}{\sqrt{3}}\quad\Rightarrow\quad\csc^{-1}(-\frac{2}{\sqrt{3}})=-\frac{\pi}{3}$
$(c)$
$\displaystyle \sin\frac{\pi}{6}=\frac{1}{2}\quad\Rightarrow\quad \csc(\frac{\pi}{6})=\frac{2}{\sqrt{3}}\quad\Rightarrow\quad\csc^{-1}(\frac{2}{1})=\frac{\pi}{6}$
