Solve `sin x gt -1/2`

Explanation

We have `sin x gt -1/2`
Since `sin x` is periodic function with period `2pi`, we solve inequality for the length of the interval `2pi` and then we generalize the solution by adding `2npi, n in Z`.
Let us solve the inequality for the value of `x in [-pi/2, (3pi)/2]` (we can choose any other interval of length `2pi`)
Draw the graph of `y= sin x` and `y=-1/2` as shown in the following figure.

From above figure, `sin x gt - 1/2` when `- pi/6 lt x lt (7 pi)/6`
On generalizing above solution, we get
`2n pi-pi/6 lt x lt 2 n pi + (7pi)/6, n in Z`