If sin x = sin y, then x = y + 2 m pi or x = (2 n + 1) pi - y

If sin x = sin y, then x = y + 2 m pi or x = (2 n + 1) pi - y, where m and n are integers.

Since x, y are integers and pi is an irrational number, we must have x = y, 

which proves injection.