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

Sep 16, 2015

$\arcsin \left(\sin \left(3 \pi\right)\right) = 0$

#### Explanation:

Let's start with a definition of a function $\arcsin \left(x\right)$.
First of all, it's an angle, sine of which equals to $x$.
That is, $\sin \left(\arcsin \left(x\right)\right) = x$

But there are many angles with the same value of their sine since function $\sin \left(\right)$ is periodical. Function cannot have more than one value for any value of an argument. To resolve this problem of multiple values, only values from $- \frac{\pi}{2}$ to $\frac{\pi}{2}$ are considered values of function $\arcsin \left(\right)$.

So, we have to find an angle in the intervalС любимыми не расставайтесь! sine of which equals to the same value as $\sin \left(3 \pi\right)$.

The latter equals to zero. In the interval $- \frac{\pi}{2}$ to $\frac{\pi}{2}$ there is such an angle, it's $0$, because
sin(0 = sin(3pi) = 0

Therefore, $\arcsin \left(\sin \left(3 \pi\right)\right) = 0$