if `a=cos 2 and b =sin7,` then

Options

A. `a gt0, b gt0`
B. `ab lt0`
C. `a gtb`
D. `a ltb`

Explanation

To solve the problem given that \( a = \cos(2) \) and \( b = \sin(7) \), we need to analyze the values of \( a \) and \( b \) based on their angles in radians.

### Step-by-Step Solution:

1. **Identify the Quadrants**:
- The angle \( 2 \) radians is between \( \frac{\pi}{2} \) (approximately \( 1.57 \)) and \( \pi \) (approximately \( 3.14 \)). Therefore, \( 2 \) radians lies in the **second quadrant**.
- The angle \( 7 \) radians is greater than \( 2\pi \) (approximately \( 6.28 \)) but less than \( \frac{5\pi}{2} \) (approximately \( 7.85 \)). Thus, \( 7 \) radians lies in the **third quadrant**.

2. **Determine the Signs of Trigonometric Functions**:
- In the **second quadrant**, the cosine function is negative. Therefore, \( a = \cos(2) < 0 \).
- In the **third quadrant**, the sine function is also negative. Thus, \( b = \sin(7) < 0 \).

3. **Evaluate the Product \( ab \)**:
- Since both \( a \) and \( b \) are negative, their product \( ab \) will be positive:
\[ ab > 0 \]

4. **Conclusion**:
- We conclude that \( ab > 0 \).