How to solve $1/2 \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0$?

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How to solve trigonomtry function involving $\sin x \cos x$ and $\sin 2x$:

$$\frac{1}{2} \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0. $$

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1 Answer

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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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