Question
In which quadrant would 0 be if sin0 = 1/2 and tan 0 > 0
Answers
GPT-4o mini
To determine in which quadrant the angle \( \theta \) lies given that \( \sin \theta = \frac{1}{2} \) and \( \tan \theta > 0 \), we can analyze the information:
1. **Sine Value**: The equation \( \sin \theta = \frac{1}{2} \) corresponds to specific angle values. In the standard unit circle, \( \sin \theta = \frac{1}{2} \) occurs at:
- \( \theta = 30^\circ \) (or \( \frac{\pi}{6} \) radians) in the first quadrant
- \( \theta = 150^\circ \) (or \( \frac{5\pi}{6} \) radians) in the second quadrant
2. **Tangent Condition**: The condition \( \tan \theta > 0 \) indicates that the sine and cosine of the angle must be either both positive (first quadrant) or both negative (third quadrant).
- In the first quadrant, both sine and cosine are positive, resulting in \( \tan \theta > 0 \).
- In the second quadrant, sine is positive but cosine is negative, which results in \( \tan \theta < 0 \).
Since \( \tan \theta > 0 \), the angle must be in the first quadrant.
Therefore, the correct quadrant for \( \theta \) is the **first quadrant**.
1. **Sine Value**: The equation \( \sin \theta = \frac{1}{2} \) corresponds to specific angle values. In the standard unit circle, \( \sin \theta = \frac{1}{2} \) occurs at:
- \( \theta = 30^\circ \) (or \( \frac{\pi}{6} \) radians) in the first quadrant
- \( \theta = 150^\circ \) (or \( \frac{5\pi}{6} \) radians) in the second quadrant
2. **Tangent Condition**: The condition \( \tan \theta > 0 \) indicates that the sine and cosine of the angle must be either both positive (first quadrant) or both negative (third quadrant).
- In the first quadrant, both sine and cosine are positive, resulting in \( \tan \theta > 0 \).
- In the second quadrant, sine is positive but cosine is negative, which results in \( \tan \theta < 0 \).
Since \( \tan \theta > 0 \), the angle must be in the first quadrant.
Therefore, the correct quadrant for \( \theta \) is the **first quadrant**.