# How do you find the exact value of arcsin(sin(2))?

Apr 29, 2018

The exact value of $\arcsin \left(\sin \left(2\right)\right)$ is simply $2$.

#### Explanation:

Whenever we take the arcsin of sin, or the arccos of cos, or the inverse of any trig function, they always cancel each other out. So $\arctan \left(\tan \left(3\right)\right) = 3$, $\arcsin \left(\sin \left(1\right)\right) = 1$, and so on.

The reason for this is because to find the arcsin of a given number, for instance, $\arcsin x$, we are basically asking "When will the sin of some number equal $x$?"

So with this problem, instead of $x$ we have the arcsin of sin. Thus we are basically asking "When will the sin of some number equal $\sin \left(2\right)$?"

As an equation with our desired answer being n, that looks like

$\sin \left(n\right) = \sin \left(2\right)$

So $n$ is clearly $2$.

This also works no matter order we're taking $\sin$ and $\arcsin$ in. So for example,

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