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Is it generally true that $|\cos(z)|\leq1$, $|\sin(z)|\leq1$ $\forall z \in \mathbb{C}$? I think I'm missing something here (I think it does not hold, only if $z \in \mathbb{R}$). If this were not the case, then could someone give me a concrete example of this?

Thanks for the help, this question has been troubling my mind for a while.

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  • $\begingroup$ By Liouville's theorem, bounded entire functions are constant. If what you ask was true, $\sin$ and $\cos$ would be bounded entire functions and thus constant, which is absurd. $\endgroup$ – PhoemueX Oct 6 '15 at 18:25
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Take $z = it, t\in\Bbb R$. $\lim_{t\to\infty}\sin(it)=\cdots$

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No, $\cos$ and $\sin$ are surjective functions. We have $\cos(\mathbb C)=\sin(\mathbb C)=\mathbb C$. If you have a holomorphic function on $\mathbb C$ which is not constant then it is almost surjective meaning that you have $f(\mathbb C)=A$ where $A=\mathbb C$ or $A=\mathbb C\setminus\{a\}$ for one $a\in\mathbb C$.

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