Solutions of sin x equals sin a

Theorem
Let $\alpha \in \closedint {-1} 1$ be fixed.

Let:
 * $(1): \quad \sin x = \sin \alpha$

The solution set of $(1)$ is:


 * $\set {x \in \R: \forall n \in \Z: x = n \pi + \paren {-1}^n \alpha}$