Question

What is the value of ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$?

Answer 1

Let $y={\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$

$\sin y=\sin \frac{2\pi }{3}$

$\sin y=\sin \left({120}^o\right)$

But range of principal value of ${\sin }^{-1}$ is $[\frac{-\pi }{2},\frac{\pi }{2}]$

i.e., $[-{90}^o,{90}^o]$

Hence $y={120}^o$ is not possible

Now,

$\sin y=\sin {120}^o$

$\sin y=\sin ({180}^o-{60}^o)$

$\sin y=\sin {60}^o$ $[\because \sin (180-\theta )=\sin \theta ]$

$\sin y=\sin (60\times \pi /180)$

$\sin y=\sin \pi /3$

Hence $y=\pi /3$

Which is the range of principal value of ${\sin }^{-1}$ i.e., $[-\pi /2.\pi /2]$

Hence,

${\sin }^{-1}\left(\sin 2\pi /3\right)=y=\frac{\pi }{3}$.