If a line makes an angle of `pi/4` with the positive direction of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is a. `pi/3` b. `pi/4` c. `pi/2` d. `pi/6`

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. `(pi)/(6)`

Explanation

To solve the problem, we need to find the angle that a line makes with the positive direction of the z-axis, given that it makes an angle of \( \frac{\pi}{4} \) with both the x-axis and y-axis.

### Step-by-Step Solution:

1. **Understand Direction Cosines**:
- Let \( \alpha \) be the angle with the x-axis, \( \beta \) be the angle with the y-axis, and \( \gamma \) be the angle with the z-axis.
- The direction cosines are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \]

2. **Given Angles**:
- From the problem, we know:
\[ \alpha = \frac{\pi}{4}, \quad \beta = \frac{\pi}{4} \]

3. **Use the Relation of Direction Cosines**:
- The relation between the direction cosines is:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

4. **Substitute the Values**:
- Substitute \( \alpha \) and \( \beta \) into the equation:
\[ \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1 \]

5. **Calculate \( \cos \left(\frac{\pi}{4}\right) \)**:
- We know that:
\[ \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
- Therefore:
\[ \cos^2 \left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

6. **Substitute Back into the Equation**:
- Now substituting back:
\[ \frac{1}{2} + \frac{1}{2} + \cos^2 \gamma = 1 \]
- This simplifies to:
\[ 1 + \cos^2 \gamma = 1 \]

7. **Solve for \( \cos^2 \gamma \)**:
- Subtract 1 from both sides:
\[ \cos^2 \gamma = 0 \]

8. **Find \( \gamma \)**:
- Taking the square root:
\[ \cos \gamma = 0 \]
- The angle \( \gamma \) for which \( \cos \gamma = 0 \) is:
\[ \gamma = \frac{\pi}{2} \]

9. **Conclusion**:
- Therefore, the angle that the line makes with the positive direction of the z-axis is:
\[ \boxed{\frac{\pi}{2}} \]