$\begin{array}{l} \sin x = \dfrac{2}{3}\\ \Leftrightarrow \left[ \begin{array}{l} x = \arcsin \left( {\dfrac{2}{3}} \right) + k2\pi \\ x = \pi - \arcsin \left( {\dfrac{2}{3}} \right) + k2\pi \end{array} \right.\\ x \in \left( { - \pi ;\pi } \right) \Rightarrow \left[ \begin{array}{l} - \pi < \arcsin \left( {\dfrac{2}{3}} \right) + k2\pi < \pi \\ - \pi < \pi - \arcsin \left( {\dfrac{2}{3}} \right) + k2\pi < \pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\\ k = 0 \Rightarrow x = \arcsin \left( {\dfrac{2}{3}} \right),x = \pi - \arcsin \left( {\dfrac{2}{3}} \right) \in \left( { - \pi ;\pi } \right)\\ k = 1 \Rightarrow x = \arcsin \left( {\dfrac{2}{3}} \right) + k2\pi ,3\pi - \arcsin \left( {\dfrac{2}{3}} \right) \notin \left( { - \pi ;\pi } \right)\\ k = - 1 \Rightarrow x = \arcsin \left( {\dfrac{2}{3}} \right) - 2\pi ,x = - \pi - \arcsin \left( {\dfrac{2}{3}} \right) \notin \left( { - \pi ;\pi } \right) \end{array}$
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