Write the value of ^ -1 (- 3 2 )+ ^ -1 (-1 2 ) - Vidyadip

Question

Write the value of $\sin^{-1}\Big(\frac{-\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{-1}{2}\Big)$

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Answer

$\sin^{-1}(-\text{x})=\sin^{-1}\text{x},\text{x}\in[-1,1]$
$\cos^{-1}(-\text{x})=\pi-\cos^{-1}\text{x},\text{x}\in[-1,1]$
$\therefore\ \sin^{-1}\Big(\frac{-\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{-1}{2}\Big)$
$=-\sin^{-1}\Big(\frac{\sqrt3}{2}\Big)+\pi-\cos^{-1}\Big(\frac{1}{2}\Big)$
$=-\sin^{-1}\Big(\sin\frac{\pi}{3}\Big)+\pi-\cos^{-1}\Big(\cos\frac{\pi}{3}\Big)$
$=-\frac{\pi}{3}+\pi-\frac{\pi}{3}$
$=\frac{\pi}{3}$
$\therefore\ \sin^{-1}\Big(\frac{-\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{-1}{2}\Big)=\frac{\pi}{3}$

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