8.049. $\cos 3 x=2 \sin \left(\frac{3 \pi}{2}+x\right)$

Translate the above text into English, preserving the original text's line breaks and format, and output the translation result directly.

8.049. $\cos 3 x=2 \sin \left(\frac{3 \pi}{2}+x\right)$
## Solution.

Let's rewrite the given equation as

$$
\begin{aligned}
& 4 \cos ^{3} x-3 \cos x=-2 \cos x, \Leftrightarrow 4 \cos ^{3} x-\cos x=0 \Leftrightarrow \\
& \Leftrightarrow \cos x\left(4 \cos ^{2} x-1\right)=0
\end{aligned}
$$

## From this

1) $\cos x=0, \quad x_{1}=\frac{\pi}{2}+\pi k=\frac{\pi}{2}(2 k+1), k \in Z$;
2) $4 \cos ^{2} x-1=0, \quad \cos x= \pm \frac{1}{2}, \quad x_{2}= \pm \frac{\pi}{3}+\pi n, \quad n \in Z$.

Answer: $\quad x_{1}=\frac{\pi}{2}(2 k+1) x_{2}= \pm \frac{\pi}{3}+\pi n, \quad k, n \in Z$.