Question

How do you evaluate `arcsin(sin3pi)` without a calculator?

Answer 1

`0`

Explanation:

`sin x` goes through zero for `x=...-2pi, -pi, 0, pi, 2pi, 3pi...`.
So `arcsin( sin 3pi))=arcsin(0)`.
By convention, the principal value for `arcsin(x)` is taken as the unique value `theta` in the range `-pi/2<=theta<=+pi/2` such that `sin(theta)=x`. In this case that value is 0