Let $z=Re^{i\alpha}$ where $\alpha$ is fixed and $0<\alpha<\pi/2$. Prove that $|e^{-iz}|, |\cos z|$ and $|\sinh z|$ tend. to infinity as $R \to \infty$
I'd really need some help with this problem. So far, I've been only able to prove that the three quanties are less or equal than some other quantity that tends to infinity as $R \to \infty$ by using the power series representations or De Moivre's formula. But that doesn't solve the problem.
I think that Liouville's Theorem would work by contradiction since the functions are holomorphic and definitely not constant, but I'd like to work it out without invoking Liouville's. Thanks in advance!
By using Euler's identity $$\{Re^{i\alpha}\mid \alpha\in\{0,\pi/2\}=\{R(\cos(\alpha)+i\sin(\alpha))\mid \alpha\in\{0,\pi/2\}\}.$$ By the choice of quadrant $\cos(\alpha),\sin(\alpha)>0$. So one has $$|e^{-iz}|=e^{Re(-iz)}=e^{R\sin(\alpha)}\overset{R \rightarrow \infty}\longrightarrow \infty.$$ And similarly for the others, using the reverse triangle inequality. For example the second $$|\cos(z)|=\left|\frac{e^{iz}-e^{-iz}}{2}\right|\ge \frac{|e^{-iz}|-|e^{iz}|}{2}\overset{R \rightarrow \infty}\longrightarrow \infty$$