Complex Sine Function is Unbounded

Theorem

Let $\sin: \C \to \C$ be the complex sine function.

Then $\sin$ is unbounded.

Proof

By Complex Sine Function is Entire, we have that $\sin$ is an entire function.

Aiming for a contradiction, suppose that $\sin$ was a bounded function.

Then, by Liouville's Theorem, we would have that $\sin$ is a constant function.

However we have, for instance, by Sine of Zero is Zero:

$\sin 0 = 0$

and by Sine of 90 Degrees:

$\sin \dfrac \pi 2 = 1$

Therefore, $\sin$ is clearly not a constant function, a contradiction.

We hence conclude, by Proof by Contradiction, that $\sin$ is unbounded.

$\blacksquare$

Also see

• Complex Cosine Function is Unbounded