Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question

Question: What is the maximum value of sine function taking domain as $\Bbb C$?

My answer is: The maximum value is not defined.

Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ is not well ordered, hence the maximum value of sine function can not be defined.

Answer 1

Your reasoning is clearly wrong. You could say the same thing for practically all nonreal-valued function. When someone is asking for the maximum of a nonreal-valued object, it is implied that you have to consider the absolute value (or, more generally, some kind of norm).

The proof that $\sin z$ is unbounded on $\mathbb C$ can be traced as follows (there might be some wrong sign! Check it!). By definition,

$$\sin z = \frac{e^{iz} - e^{-iz}}{2i}.$$

Now, set $z = x + iy$ (with $x,y \in \mathbb R$):

$$e^{iz} = e^{ix - y}, \quad e^{-iz} = e^{-ix + y}.$$

Hence, using the De Moivre formula and making some simple algebraic manipulations,

$$\sin z = -i \cos x \sinh y + \sin x \cosh y.$$

So we have

$$\vert {\sin z} \vert^2 = \sinh^2 y + \sin^2 x. $$

Hyperbolic sine is obviously unbounded on $\mathbb R$, while sine is bounded, hence also $\sin z$ is unbounded on $\mathbb C$.

Answer 2

Hint: Since sinz is entire hence by Picard's little theorem, it must assume all values on complex plane C with at most one possible exception.