In which quadrant does \( \theta \) lie given \( \tan \theta>0 \) and \( \sin \theta<0 \) ?

To determine which quadrant \( \theta \) lies in based on the conditions provided, we need to analyze the signs of the trigonometric functions. Since \( \tan \theta > 0 \), this means that both sine and cosine must have the same sign, which can be either positive or negative. The tangent is positive in the first and third quadrants. However, the second condition states that \( \sin \theta < 0 \), meaning sine must be negative. This is true in the third and fourth quadrants. Combining these two conditions, we conclude that \( \theta \) lies in the third quadrant, where both sine is negative, and tangent is positive. So, the answer is that \( \theta \) is in the third quadrant.