# How do you evaluate arcsin(sin((7pi)/3))?

Apr 13, 2016

$\arcsin \left(\sin \left(\frac{7 \pi}{3}\right)\right) = \left(\frac{\pi}{3}\right)$

#### Explanation:

If $\arcsin x = \theta$

then $\sin \theta = x$.

Let $\theta = \left(\frac{7 \pi}{3}\right)$ and $\sin \left(\frac{7 \pi}{3}\right) = x$

Then $\arcsin \left(\sin \left(\frac{7 \pi}{3}\right)\right) = \arcsin x = \theta = \frac{7 \pi}{3}$

However, in case of trigonometric ratios, for each $x$, their could be multiple vales of $\theta$ (such as $2 n \pi + \theta$) e.g. not only $\sin \left(\frac{7 \pi}{3}\right) = x$, but so is for $\sin \left(\frac{\pi}{3}\right)$, $\sin \left(- 5 \frac{\pi}{3}\right)$, $\sin \left(13 \frac{\pi}{3}\right)$ etc.

Hence, the range for $\theta = \arcsin x$ is limited to $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$

Hence $\arcsin \left(\sin \left(\frac{7 \pi}{3}\right)\right) = \left(\frac{\pi}{3}\right)$