By considering the graph of $y=\sin x$ prove that, for $0\le x\le \pi /2$,

${\displaystyle \frac{2x}{\pi }\le \sin x\le x.}$

By considering the graph of $y=\tan x$ prove that, for $0<a<b<\pi /2$,

${\displaystyle \frac{\tan a}{\tan b}<\frac{a}{b}.}$

You can use the excellent shareware Graphmatica (downloadable from http://nrich.maths.org/downloads/graphmatica.zip ), or a graphic calculator, to experiment with the graphs here.