\(\Leftrightarrow sin\left(x+\frac{\pi}{3}\right)=-cos\left(\frac{\pi}{5}\right)\)
\(\Leftrightarrow sin\left(x+\frac{\pi}{3}\right)=sin\left(\frac{\pi}{5}-\frac{\pi}{2}\right)\)
\(\Leftrightarrow sin\left(x+\frac{\pi}{3}\right)=sin\left(-\frac{3\pi}{10}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{3}=-\frac{3\pi}{10}+k2\pi\\x+\frac{\pi}{3}=\frac{13\pi}{10}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{19\pi}{30}+k2\pi\\x=\frac{29\pi}{30}+k2\pi\end{matrix}\right.\)