Finding the norm of a complex trigonometric function?

Question

Given that the complex norm $|z| = 1$, how would I go about proving that $|cos(z)| \leq e$? Just a hint would be helpful.

Answer 1

$z=e^{i\theta}=\cos\theta+i\sin\theta$.

$\cos z=\frac12(e^{ie^{i\theta}}+e^{-ie^{i\theta}})=\ldots$

Answer 2

Write out $$\cos{z} = \frac{e^{iz} + e^{-iz}}{2} $$ and apply the triangle inequality. There will be some minor details to work out, but they should be fairly straightforward :)