Answer
(a) $\arcsin(-1)=-\frac{\pi}{2}$
(b) $\arcsin\Big(-\frac{1}{\sqrt2}\Big)=-\frac{\pi}{4}$
Work Step by Step
*Recall the definition of arcsine:
$y=\sin^{-1}x$ is the number in $[-\pi/2,\pi/2]$ for which $\sin y=x$
(a) $$\arcsin(-1)=a$$
According to the defintion: $$\sin a=-1\hspace{1cm}\text{and}\hspace{1cm}a\in[-\pi/2,\pi/2]$$
So $a=-\frac{\pi}{2}$. In other words, $$\arcsin(-1)=-\frac{\pi}{2}$$
(b) $$\arcsin\Big(-\frac{1}{\sqrt2}\Big)=\arcsin\Big(-\frac{\sqrt2}{2}\Big)=a$$
According to the defintion: $$\sin a=-\frac{\sqrt2}{2}\hspace{1cm}\text{and}\hspace{1cm}a\in[-\pi/2,\pi/2]$$
So $a=-\frac{\pi}{4}$. In other words, $$\arcsin\Big(-\frac{1}{\sqrt2}\Big)=-\frac{\pi}{4}$$
