# How to find the general solution of $\sin \left(x+\frac{2\pi }{3}\right)=3\sin \left(x+\frac{\pi }{3}\right)$

Find the general solution of the equation.

\begin{eqnarray} \sin \left(x+\frac{2\pi }{3}\right)=3\sin \left(x+\frac{\pi }{3}\right)\\ \end{eqnarray}

The answers in my textbook are $n\pi -\tan ^{-1}\left(\sqrt{\frac{3}{2}}\right)$.

\begin{eqnarray} \\\sin \left(x+\frac{2\pi }{3}\right)=3\sin \left(x+\frac{\pi }{3}\right)\\ \\cos \left(x+\frac{\pi }{6}\right)=3\cos \left(x-\frac{\pi }{6}\right)\\ \end{eqnarray}

What should I do?

Would you mind telling me the method for solving this question?

Thank you for your attention.

$$\frac32\sin x+\frac{3\sqrt3}2\cos x=3\sin\left(x+\frac\pi3\right)=\sin\left(x+\frac{2\pi}3\right)=-\frac12\sin x+\frac{\sqrt3}2\cos x\implies$$
$$3\tan x+3\sqrt3=-\tan x+\sqrt3\iff4\tan x=-2\sqrt3\iff\tan x=-\frac{\sqrt3}2\ldots$$