Question
Determine the quadrant of \theta if \tan \theta = -(√(3))/(3) and \cos \theta = -(√(3))/(2).
A. Quadrant IV
B. Quadrant II
C. Quadrant III
D. Quadrant I
A. Quadrant IV
B. Quadrant II
C. Quadrant III
D. Quadrant I
Answers
GPT-4o mini
To determine the quadrant of \(\theta\) given that \(\tan \theta = -\frac{\sqrt{3}}{3}\) and \(\cos \theta = -\frac{\sqrt{3}}{2}\), we can analyze the signs of the trigonometric functions:
1. **Tangent**: \(\tan \theta = -\frac{\sqrt{3}}{3}\)
- Tangent is negative when either sine is negative and cosine is positive (Quadrant IV) or when both sine and cosine are negative (Quadrant II).
2. **Cosine**: \(\cos \theta = -\frac{\sqrt{3}}{2}\)
- Cosine is negative in Quadrant II and Quadrant III.
Now, we combine this information:
- Since \(\tan \theta\) is negative and \(\cos \theta\) is negative, the only quadrant that satisfies both conditions is **Quadrant II**. In Quadrant II, sine is positive (making tangent negative when cos is also negative).
Hence, the correct answer is:
**B. Quadrant II**
1. **Tangent**: \(\tan \theta = -\frac{\sqrt{3}}{3}\)
- Tangent is negative when either sine is negative and cosine is positive (Quadrant IV) or when both sine and cosine are negative (Quadrant II).
2. **Cosine**: \(\cos \theta = -\frac{\sqrt{3}}{2}\)
- Cosine is negative in Quadrant II and Quadrant III.
Now, we combine this information:
- Since \(\tan \theta\) is negative and \(\cos \theta\) is negative, the only quadrant that satisfies both conditions is **Quadrant II**. In Quadrant II, sine is positive (making tangent negative when cos is also negative).
Hence, the correct answer is:
**B. Quadrant II**