$\begin{array}{l} \sin \left( {\dfrac{{x + \pi }}{5}} \right) = - \dfrac{1}{2}\\ \Leftrightarrow \left[ \begin{array}{l} \dfrac{{x + \pi }}{5} = - \dfrac{\pi }{6} + k2\pi \\ \dfrac{{x + \pi }}{5} = \dfrac{{7\pi }}{6} + k2\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x + \pi = - \dfrac{{5\pi }}{6} + k10\pi \\ x + \pi = \dfrac{{35\pi }}{6} + k10\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = - \dfrac{{11\pi }}{6} + k10\pi \\ x = \dfrac{{29\pi }}{6} + k10\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}$
