Pregunta
Identify the quadrant(s) of an angle \( \theta \) that satisfies the given conditions. \( \sin \theta>0, \cos \theta>0 \) Choose the correct answer below. A. I and IV B. I and II C. I D. I and III
Ask by Vargas Mitchell. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The angle \( \theta \) is in **Quadrant I**.
**Correct Answer:**
**C. I**
Solución
To determine the quadrant(s) where an angle \( \theta \) satisfies \( \sin \theta > 0 \) and \( \cos \theta > 0 \), let's analyze the signs of sine and cosine in each quadrant:
1. **Quadrant I:**
- \( \sin \theta > 0 \)
- \( \cos \theta > 0 \)
2. **Quadrant II:**
- \( \sin \theta > 0 \)
- \( \cos \theta < 0 \)
3. **Quadrant III:**
- \( \sin \theta < 0 \)
- \( \cos \theta < 0 \)
4. **Quadrant IV:**
- \( \sin \theta < 0 \)
- \( \cos \theta > 0 \)
Given the conditions \( \sin \theta > 0 \) and \( \cos \theta > 0 \), only **Quadrant I** satisfies both.
**Correct Answer:**
**C. I**
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Simplifique esta solución The Deep Dive
The angle \( \theta \) must be in the first quadrant to satisfy both conditions that \( \sin \theta > 0 \) and \( \cos \theta > 0 \). In the first quadrant (0 to 90 degrees), both sine and cosine values are positive. Hence, the correct answer is C. I. Now, if you’re ever in doubt about quadrants, remember this nifty tip: Just think of the unit circle! In the first quadrant, both x (cosine) and y (sine) coordinates are positive, making it the happy zone for angles!
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