\(\Leftrightarrow sin2x=sin\left(x+\frac{\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=x+\frac{\pi}{3}+k2\pi\\2x=\frac{2\pi}{3}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{3}+k2\pi\\x=\frac{2\pi}{9}+\frac{k2\pi}{3}\end{matrix}\right.\)
\(0< x< 2\pi\Rightarrow\left[{}\begin{matrix}0< \frac{\pi}{3}+k2\pi< 2\pi\\0< \frac{2\pi}{9}+\frac{k2\pi}{3}< 2\pi\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}-\frac{1}{6}< k< \frac{5}{6}\\-\frac{1}{3}< k< \frac{8}{3}\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}k=0\\k=\left\{0;1;2\right\}\end{matrix}\right.\) \(\Rightarrow x=\left\{\frac{\pi}{3};\frac{2\pi}{9};\frac{8\pi}{9};\frac{14\pi}{9}\right\}\)