Prove trigonometric inequality Could anyone help me prove the following inequality for $x>0$$$x(2+\cos x)>3\sin x$$
If you could just show me the first few steps, that would be great.

Hint: 
First, since $2+\cos x\geq 1$, we have 
$$x(2+\cos x)\geq x>3\geq3\sin x \hspace{10pt} \forall x>3$$
So it remains to prove the inequality for $0<x<3$ (clearly it is true for $x=3$).
Consider the function $f(x)=x(2+\cos x)-3\sin x$. Then 
$$f'(x)=2+\cos x-x\sin x-3\cos x=2-2\cos x-x\sin x$$
Check that $f(x)$ is positive for $0<x<3$ by finding the $\min/\max$ of $f(x)$ and use your knowledge that $f(x)>0$ for all $x\geq 3$.
