Determine the quadrant of an angle given the signs of its sine and tangent.

1. Analyze the given conditions: We are given two conditions: $\sin \theta < 0$ and $\tan \theta > 0$. We need to determine the quadrant in which $\theta$ lies based on these conditions.

1. Determine the quadrant for $\sin \theta < 0$: The sine function represents the y-coordinate on the unit circle. The y-coordinate is negative in Quadrant III and Quadrant IV.

1. Determine the quadrant for $\tan \theta > 0$: The tangent function is the ratio of the sine to the cosine ($\tan \theta = \frac{\sin \theta}{\cos \theta}$). The tangent is positive when both sine and cosine have the same sign. This occurs in Quadrant I (where both are positive) and Quadrant III (where both are negative).

4. Find the common quadrant: We need to find the quadrant that satisfies both conditions.
- Condition 1 ($\sin \theta < 0$) is true in Quadrant III and Quadrant IV.
- Condition 2 ($\tan \theta > 0$) is true in Quadrant I and Quadrant III.
The only quadrant that satisfies both conditions is Quadrant III.