Question

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

Answer 1

The exact value of `arcsin(sin(2))` 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(tan(3))=3`, `arcsin(sin(1))=1`, and so on.

The reason for this is because to find the arcsin of a given number, for instance, `arcsinx`, 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(2)`?"

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

`sin(n)=sin(2)`

So `n` is clearly `2`.

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

`sin(arcsin(0))=0`