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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 $x + y + z = \pi$?

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up vote 2 down vote accepted

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.

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