Challenge Level

By considering the graph of $y=\sin x$ prove that, for $0\leq x \leq \pi/2$, $${2x\over \pi} \leq \sin x \leq x.$$ By considering the graph of $y=\tan x$ prove that, for $0 < a < b < \pi/2$, $${\tan a \over \tan b} < {a\over b}.$$

Can you find similar inequalities which hold for different ranges of $x$?