How to prove that a trigonometric function is positive in an interval?

Question

How can I prove that the function $f(x)=\sin(x)-x\cos(x)$ is positive in the interval $]0,π]$?

Answer 1

We have that

$$f(x)=\sin x-x\cos x\implies f(0)=0,\, f(\pi)=\pi$$

and since $\sin x > 0$ for $x\in(0,\pi)$

$$f'(x)=x\sin x>0$$

thus $f(x)$ is strictly increasing on that interval and $f(x)>0$.