If cosx=a, cosw=-a, and `(pi)/(2)ltxlt0`, which of the following is positive value of w?

Options

A. `pi-x`
B. `x-pi`
C. `2pi-x`
D. `x+2pi`

Answer

To solve the problem, we need to find the positive value of \( w \) given that \( \cos x = a \), \( \cos w = -a \), and \( \frac{\pi}{2} < x < 0 \).

### Step-by-Step Solution:

1. **Understanding the Range of \( x \)**:
Since \( \frac{\pi}{2} < x < 0 \), we note that \( x \) is in the fourth quadrant. In this quadrant, the cosine function is positive. Therefore, \( \cos x = a \) indicates that \( a \) is a positive value.

**Hint**: Remember that in the fourth quadrant, cosine values are positive.

2. **Finding the Quadrant for \( w \)**:
Given that \( \cos w = -a \) and \( a \) is positive, it follows that \( \cos w \) must be negative. The cosine function is negative in the second and third quadrants.

**Hint**: Cosine is negative in the second and third quadrants.

3. **Determining the Quadrant for \( w \)**:
Since \( x \) is in the fourth quadrant, we can infer that \( w \) must be in the second quadrant because the third quadrant would not yield a positive \( w \).

**Hint**: Identify which quadrants correspond to the signs of the cosine function.

4. **Finding the Relationship Between \( w \) and \( x \)**:
In the second quadrant, the cosine of an angle is related to the cosine of its reference angle. The reference angle for \( w \) can be expressed as:
\[ w = \pi - x \]
This relationship holds because in the second quadrant, the cosine of \( w \) is equal to the negative cosine of \( x \).

**Hint**: Use the property of angles in different quadrants to relate \( w \) and \( x \).

5. **Conclusion**:
Therefore, the positive value of \( w \) is:
\[ w = \pi - x \]

### Final Answer:
The positive value of \( w \) is \( \pi - x \).