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# Find the values of each of the expressions in Exercises 16 to 18. 16. sin inverse (sin 2pi / 3)

Find the values of each of the expressions in Exercises 16 to 18.

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

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Given $\sin^{-1}\left (\sin\frac{2\pi}{3} \right )$;

We know that $\sin^{-1}(\sin x) = x$

If the value of x belongs to  $\left [ \frac{-\pi}{2},\frac{\pi}{2} \right ]$ then we get the principal values of $\sin^{-1}x$.

Here, $\frac{2\pi}{3} \notin \left [ \frac{-\pi}{2},\frac{\pi}{2} \right ]$

We can write $\sin^{-1}\left (\sin\frac{2\pi}{3} \right )$  is as:

$\sin^{-1}\left [ \sin\left ( \pi-\frac{2\pi}{3} \right ) \right ]$

= $\sin^{-1}\left [ \sin \frac{\pi}{3} \right ]$  where $\frac{\pi}{3} \epsilon \left [ \frac{-\pi}{2}, \frac{\pi}{2} \right ]$

$\therefore \sin^{-1}\left (\sin\frac{2\pi}{3} \right )=\sin^{-1}\left [ \sin \frac{\pi}{3} \right ]=\frac{\pi}{3}$

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