Trigonometric bounds Is there a nice way to show:
$\sin(x) + \sin(y) + \sin(z) \geq 2$ for all $x,y,z$ such that $0 \leq x,y,z \leq \frac{\pi}{2}$ and $x + y + z = \pi$?

Use the following inequality: $$\sin(x) \geq x\frac{2}{\pi} , x \in [0,\pi/2]$$
And to prove this inequality, Consider the function:
$ f(x) = \frac{\sin(x)}{x} $ if $x \in (0, \pi/2]$ and $f(x) = 1$ if $x=0$. Now show $f$ decreases on $[0,\pi/2]$. Hint: Use Mean Value Theorem.
