Real Sine Function is neither Injective nor Surjective

Theorem
The real sine function is neither an injection nor a surjection.

Proof
This is immediately apparent from the graph of the sine function:

For example:
 * $\map \sin 0 = \map \sin \pi = 0$

and so the real sine function is not an injection.

Then, for example:
 * $\nexists x \in \R: \map \sin x = 2$

and so the real sine function is not a surjection.