Question

In which quadrant would 0 be if sin 0 = 1/2 and tan 0 > 0?

Answer 1

To find the quadrant for the angle \( \theta \) where \( \sin \theta = \frac{1}{2} \) and \( \tan \theta > 0 \), let's analyze the conditions.

1. **For \( \sin \theta = \frac{1}{2} \)**: The sine function is positive in the first and second quadrants. The specific angles where \( \sin \theta = \frac{1}{2} \) are \( \theta = 30^\circ \) (or \( \frac{\pi}{6} \)) in the first quadrant and \( \theta = 150^\circ \) (or \( \frac{5\pi}{6} \)) in the second quadrant.

2. **For \( \tan \theta > 0 \)**: The tangent function is positive in the first and third quadrants. Thus, if the tangent is greater than zero, the angle must either be in the first quadrant or in the third quadrant.

Combining both conditions:
- The first quadrant satisfies that \( \sin \theta = \frac{1}{2} \) and \( \tan \theta > 0 \).
- The second quadrant satisfies that \( \sin \theta = \frac{1}{2} \) but \( \tan \theta < 0 \).
- The third quadrant does not satisfy the sine condition.

Therefore, the only quadrant where both conditions hold true is the **first quadrant**. Hence, \( \theta \) is in the first quadrant.