Every Angle for a Value

To find every angle in one turn with a given value, run the method backwards. The size names the reference angle; the sign names two quadrants.

Take $\sin\left(\theta\right) = \frac{-\sqrt{2}}{2}$. The size √2/2 means reference angle 45°. Sine is negative in quadrants III and IV. In III the angle is 180° + 45° = 225°; in IV it is 360° − 45° = 315°. So two angles, one in each quadrant where the sign fits.

The usual slip is to stop after one angle. Every value strictly between −1 and 1 occurs twice in a turn.