Find the principal values: ^ -1 (-1 2 ) - Vidyadip

Question

Find the principal values:
$\sin^{-1}\bigg(-\frac{1}{2}\bigg)$

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Answer

$\text{Let y}= \sin^{-1}\left(\frac{-1}{2}\right)\text {where}-\frac{\pi}{2}\leq\text{Y}\leq\frac{\pi}{2}$
$\therefore\ \text{sin Y} =-\frac{1}{2}\ \ \ \text{where}\ -\frac{\pi}{2}\leq \text{Y}\ \leq\ \frac{\pi}{2}$
$\therefore\text{Y}=\ -\frac{\pi}{6}\ \bigg[\because\sin\bigg(-\frac{\pi}{6}\bigg)=-\sin\frac{\pi}{6}=-\frac{1}{2}\bigg]$
$\therefore$ required principal value $ =-\frac{\pi}{6}$

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