# How do you find the inverse of sin?

Apr 15, 2015

The inverse of the $\sin$ function is the $\arcsin$ function.
But sine itself, would not be invertible because it's not injective, so it's not bijective (invertible).
To obtain arcsine function we have to restrict the domain of sine to $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$.

$\arcsin \left(\sin \left(\theta\right)\right) = \theta \text{ if and only if} - \frac{\pi}{2} \le \theta \le \frac{\pi}{2}$

For example:
Since $\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$
$\arcsin \left(\frac{1}{2}\right) = \frac{\pi}{6}$

If you meant by your question "how do you calculate the value of the $\arcsin$ for some given number?"
then the answer is that there is no easy way to do this (with the exception of a few special values) just as there is no easy way to calculate the value of $\sin$ for most values.