Question:
Write the value of $\sin ^{-1}\left(\frac{-\sqrt{3}}{2}\right)+\cos ^{-1}\left(\frac{-1}{2}\right)$
Solution:
$\sin ^{-1}(-x)=-\sin ^{-1} x, x \in[-1,1]$
$\cos ^{-1}(-x)=\pi-\cos ^{-1} x, x \in[-1,1]$
$\therefore \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\cos ^{-1}\left(-\frac{1}{2}\right)$
$=-\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)+\pi-\cos ^{-1}\left(\frac{1}{2}\right)$
$=-\sin ^{-1}\left(\sin \frac{\pi}{3}\right)+\pi-\cos ^{-1}\left(\cos \frac{\pi}{3}\right)$
$=-\frac{\pi}{3}+\pi-\frac{\pi}{3}$
$=\frac{\pi}{3}$
$\therefore \sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\cos ^{-1}\left(-\frac{1}{2}\right)=\frac{\pi}{3}$
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