Question
1.4 Quostion 6, 1.4.43 HW Scores 33.33\%, 5 of 15 points Pointa: 0 of 1 Savo Identify the quadrant or quadrants for the angles satisfying the following conditions. \( \sin \alpha<0 \) and \( \cos \alpha<0 \) The angles \( \alpha \) satisfying the above conditions are in Quadrant (Type your answer as 1, 2, 3, or 4 . Use commas to separate answers if needed.)
Ask by Lawson Bush. in the United States
Feb 03,2025
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Answer
The angles \( \alpha \) are in Quadrant 3.
Solution
To determine in which quadrant(s) the angle \( \alpha \) satisfies both \( \sin \alpha < 0 \) and \( \cos \alpha < 0 \), let's analyze each quadrant:
1. **Quadrant I**:
- \( \sin \alpha > 0 \)
- \( \cos \alpha > 0 \)
- **Does not satisfy** the conditions.
2. **Quadrant II**:
- \( \sin \alpha > 0 \)
- \( \cos \alpha < 0 \)
- **Only \( \cos \alpha \) is negative**.
3. **Quadrant III**:
- \( \sin \alpha < 0 \)
- \( \cos \alpha < 0 \)
- **Both sine and cosine are negative**.
4. **Quadrant IV**:
- \( \sin \alpha < 0 \)
- \( \cos \alpha > 0 \)
- **Only \( \sin \alpha \) is negative**.
**Conclusion**: The angles \( \alpha \) that satisfy both \( \sin \alpha < 0 \) and \( \cos \alpha < 0 \) are in **Quadrant 3**.
**Answer:**
3
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Simplify this solution Extra Insights
The angles \( \alpha \) where both \( \sin \alpha < 0 \) and \( \cos \alpha < 0 \) fall in Quadrant 3. This is where the sine function is negative (as it's below the x-axis) and the cosine function is also negative (as it’s to the left of the y-axis). Remember, in Quadrant 3, angles range from 180° to 270°. When you're working with trigonometric functions, it helps to visualize the unit circle! By sketching out the circle and labeling the quadrants with the signs of sine and cosine, you’ll find it easier to remember which angles fall into which categories. Quadrant 3 will always be a mix of negative values for sine and cosine, ensuring a solid grasp on the subject!
