Proving that $\sin n\theta\leq n\sin\theta$.

Question

I am trying to prove this complex inequality. $$ \mid \Im{(z^n)}\mid\leq n\mid z^{n-1}\mid \mid\Im{(z)}\mid $$

Where $n$ is a positive integer.

This inequality is equivalent to proving: $$\sin (n\theta)\leq n\sin\theta$$ How do you prove this?

Answer 1

If the inequality is required in $[0,\pi /2]$ a simple induction argument using $\sin (n+1)\theta =\sin (n\theta) \cos \theta +\cos n \theta \sin (\theta)$ can be used.