Trigonometry inequality with 2 natural parameters.

Question

Any ideas for this trigonometry inequality? $$\sin \left({\pi\over mn } \right) \geq {1\over m} \sin \left({\pi\over n } \right)$$ where $m,n \in \Bbb N^+$. I have already tried to use induction, but I don't get anywhere. I saw a solution with derivatives, but I didn't study them.

Answer 1

You need to know that $\;0<f(x):=\sin(x)/x<1\;$ if $\;0<x<\pi.\;$ This is clear geometrically. Also $f(x)$ is monotonically decreasing and $\;\pi/(mn)\le\pi/n.\;$ Thus $\;f(\pi/(mn))\ge f(\pi/n)\;$ and the result follows.