[tex]x = \frac{\pi}{6} +2\pi k \space , \space \text{for any integer values of } k\\[/tex]
[tex]x = \frac{5\pi}{6} -2\pi k \space , \space \text{for any integer values of } k\\[/tex]
Solving with the Periodicity Identity:
[tex]\sin(x)= \frac{1}{2} \\ x = \arcsin(\frac{1}{2}) +2\pi k \\ x = \frac{\pi}{6} +2\pi k[/tex]
Solving with the Symmetry Identity:
[tex]\sin(x) = \frac{1}{2} \\ \sin(\pi -x) = \frac{1}{2} \\ \pi -x = \arcsin(\frac{1}{2}) +2\pi k \\ \pi -x = \frac{\pi}{6} +2\pi k \\ -x = \frac{\pi}{6} +2\pi k -\pi \\ -x = \frac{\pi}{6} - \frac{6\pi}{6} +2\pi k \\ -x = -\frac{5\pi}{6} + 2\pi k \\ x = \frac{5\pi}{6} -2\pi k[/tex]
