Proving inequality $|\sin(n\theta) |\le n\sin\theta$ I was trying to prove following inequality:
$$|\sin n\theta| \leq n\sin \theta \
\text{for all n=1,2,3... and } \ 
0<\theta<π $$
I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?

Let $z = \cos(θ) + i \sin(θ) $
$\displaystyle
{\sin(n\,θ)\over \sin(θ)} 
= {\Im(z^n) \over \Im(z)} 
= {z^n - 1/z^n \over z - 1/z} 
= z^{n-1} + z^{n-3}+\cdots+{1\over z^{n-3}} + {1\over z^{n-1}}$
RHS can be group in pairs (except possibly center 1, if n is odd) :
${\displaystyle z^k + {1 \over z^k} = 2 \cos(k\,x) }$
Sum of pair absolute value at most reached 2.
With RHS having n terms, $\displaystyle \left|{\sin(n\,θ)\over \sin(θ)}  \right| \le n$
