Question
Evaluate the following:
$\sin^{-1}\Big(\sin\frac{7\pi}{6}\Big)$
$\sin^{-1}\Big(\sin\frac{7\pi}{6}\Big)$
✓
Answer
We know
$\sin\big(\sin^{-1}\theta\big)=\theta$ if $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$
$\sin^{-1}\Big(\sin\frac{7\pi}{6}\Big)$
$=\sin^{-1}\Big(\sin\Big(\pi+\frac{\pi}{6}\Big)\Big)$
$=\sin^{-1}\Big(\sin\Big(-\frac{\pi}{6}\Big)\Big)$
$=-\frac{\pi}{6}$
$\sin\big(\sin^{-1}\theta\big)=\theta$ if $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$
$\sin^{-1}\Big(\sin\frac{7\pi}{6}\Big)$
$=\sin^{-1}\Big(\sin\Big(\pi+\frac{\pi}{6}\Big)\Big)$
$=\sin^{-1}\Big(\sin\Big(-\frac{\pi}{6}\Big)\Big)$
$=-\frac{\pi}{6}$
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