How do i get from:
sin x = -1/2
to: x = 7pi/6, 11pi/6

A few ways. you can enter it into a calculator for one, but the easiest way would be to reference a unit circle and look for an ordered pair where sin (the y value), is equal to -1/2

on a unit circle, the sin value is -1/2 at 7pi/6 and 11pi/6

because sin is the y value, it will additionally be in the quadrants III or IV based on the fact that (-1/2) as a sin value IS a negative and would have to be found in a quadrant in which sin is negative (the lower half of a coordinate plane, in this case)