Mathematics

The key is recognizing that both equations must be satisfied simultaneously, so you're looking for angles where the sine and tangent values match those constraints at the same time. Start by finding which quadrant(s) allow both conditions: $\sin \theta = -\frac{1}{2}$ narrows you to quadrants III and IV, while $\tan \theta = \frac{1}{\sqrt{3}} > 0$ means quadrants I or III. Only quadrant III satisfies both, so you're finding a specific angle there, then adding the full period to capture all solutions.

The key is recognizing that both equations must be satisfied simultaneously, so you're looking for angles where the sine and tangent values match those constraints at the same time. Start by finding which quadrant(s) allow both conditions: $\sin \theta = -\frac{1}{2}$ narrows you to quadrants III and IV, while $\tan \theta = \frac{1}{\sqrt{3}} > 0$ means quadrants I or III. Only quadrant III satisfies both, so you're finding a specific angle there, then adding the full period to capture all solutions.