Question

Using principal value, evaluate the following:
${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{5\pi }{6}\right)$

Answer 1

${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)+{\tan }^{-1}\left(\tan \frac{5\pi }{6}\right)$

We know that, range of principal value of $\tan$ and $\sin$ are between $(\frac{-\pi }{2},\frac{\pi }{2})$

$\Rightarrow$ ${\sin }^{-1}\left(\sin (\pi -\frac{\pi }{6})\right)+{\tan }^{-1}\left(\tan (\pi -\frac{\pi }{6})\right)$

$\Rightarrow$ ${\sin }^{-1}\left(\sin \left(\frac{\pi }{6}\right)\right)+{\tan }^{-1}\left(\tan \left(\frac{-\pi }{6}\right)\right)$

$\Rightarrow$ $\frac{\pi }{6}+\left(\frac{-\pi }{6}\right)$

$\Rightarrow$ $0$