Let \( \theta \) be an angle in standard position. Name the quadrant in which \( \theta \) lies. \[ \begin{array}{l}\tan \theta>0, \sin \theta<0 \\ \text { The angle } \theta \text { lies in which quadrant? } \\ \text { in } \\ \text { in }\end{array} \]

To determine the quadrant in which the angle \( \theta \) lies, we can analyze the given conditions: 1. \( \tan \theta > 0 \): The tangent function is positive in the first and third quadrants. 2. \( \sin \theta < 0 \): The sine function is negative in the third and fourth quadrants. Now, we need to find the quadrant where both conditions are satisfied simultaneously. - The first quadrant has \( \tan \theta > 0 \) and \( \sin \theta > 0 \) (not applicable here). - The second quadrant has \( \tan \theta < 0 \) and \( \sin \theta > 0 \) (not applicable here). - The third quadrant has \( \tan \theta > 0 \) and \( \sin \theta < 0 \) (this satisfies both conditions). - The fourth quadrant has \( \tan \theta < 0 \) and \( \sin \theta < 0 \) (not applicable here). Since the only quadrant that satisfies both conditions is the third quadrant, we conclude that: \[ \text{The angle } \theta \text{ lies in the third quadrant.} \]