<s>[INST] How to solve $1/2 \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0$? How to solve trigonomtry function involving $\sin x \cos x$ and $\sin 2x$:
$$\frac{1}{2} \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0. $$
 [/INST] Hint:
Using the identity $\sin(2x) = 2 \sin x \cos x$ we have
$$ \sin x \cos x + \sin x + 2\cos x + 2 = 0$$
Factor
$$ (1 + \cos x) \sin x + 2(1 + \cos x) = 0 \\ 
(1 + \cos x)(2 + \sin x) = 0
$$
So either $1 + \cos x = 0$ or $2 + \sin x = 0.$
Solve for $x$ in each case.
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