Question
et \( \theta \) be an angle in standard position. Name the quadrant in which \( \theta \) lies. \( \sin \theta>0, \tan \theta<0 \) The angle \( \theta \) lies in which quadrant? II III IV
Ask by Reese Rowe. in the United States
Dec 03,2024
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Tutor-Verified Answer
Answer
The angle \( \theta \) lies in Quadrant II.
Solution
To determine the quadrant in which the angle \( \theta \) lies based on the given conditions \( \sin \theta > 0 \) and \( \tan \theta < 0 \), we can analyze the signs of the trigonometric functions in each quadrant:
1. **Quadrant I**: \( \sin \theta > 0 \), \( \tan \theta > 0 \)
2. **Quadrant II**: \( \sin \theta > 0 \), \( \tan \theta < 0 \)
3. **Quadrant III**: \( \sin \theta < 0 \), \( \tan \theta > 0 \)
4. **Quadrant IV**: \( \sin \theta < 0 \), \( \tan \theta < 0 \)
From the conditions:
- \( \sin \theta > 0 \) indicates that \( \theta \) is in either Quadrant I or Quadrant II.
- \( \tan \theta < 0 \) indicates that \( \theta \) is in either Quadrant II or Quadrant IV.
The only quadrant that satisfies both conditions is **Quadrant II**.
Therefore, the angle \( \theta \) lies in **Quadrant II**.
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Simplify this solution Extra Insights
When analyzing the signs of sine and tangent, we find that \( \sin \theta > 0 \) indicates that the angle is in a quadrant where sine is positive, which is in Quadrants I and II. However, \( \tan \theta < 0 \) tells us that tangent, which is the ratio of sine to cosine, is negative, implying that sine and cosine have opposite signs; hence we are in a quadrant where sine is positive and cosine is negative. This combination of conditions occurs in Quadrant II, making it the correct quadrant for \( \theta \). So, \( \theta \) lies in Quadrant II!
