Can I represent $S = \{x: \sin(x) > 0\}$ as $\bigcup_{k\in\Bbb Z} \left[\frac {\pi}{6}+2\pi k,\frac {5}{6} \pi+2\pi k\right]$? Is it correct to represent $S = \{x: \sin(x) > 0\}$ as $\bigcup_{k\in\Bbb Z} \left[\dfrac {\pi}{6}+2\pi k,\dfrac {5}{6} \pi+2\pi k\right]$?

Draw a picture (the trigonometric circle) and you will see that $\sin$ is positive on $(0,\pi)$ so by periodicity we get
$$S=\bigcup_{k\in\Bbb Z}(2k\pi,(2k+1)\pi)$$
