If a line makes angles `(pi)/(3) and (pi)/(4)` with the x-axis and y-axis respectively, then the acute angle made by the line with z-axis is

Options

A. `pi/6`
B. `pi/4`
C. `pi/3`
D. `5pi/12`

Answer

To solve the problem, we need to find the acute angle made by a line with the z-axis, given that it makes angles of \( \frac{\pi}{3} \) with the x-axis and \( \frac{\pi}{4} \) with the y-axis.

### Step-by-step Solution:

1. **Understanding the Angles**:
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. We have:
\[ \alpha = \frac{\pi}{3}, \quad \beta = \frac{\pi}{4} \]

2. **Using the Relation**:
There is a known relation between these angles:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Calculating \( \cos^2 \alpha \)**:
First, we calculate \( \cos^2 \alpha \):
\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \quad \Rightarrow \quad \cos^2 \left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Calculating \( \cos^2 \beta \)**:
Next, we calculate \( \cos^2 \beta \):
\[ \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \cos^2 \left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

5. **Substituting Values into the Relation**:
Now, substituting \( \cos^2 \alpha \) and \( \cos^2 \beta \) into the relation:
\[ \frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1 \]

6. **Finding \( \cos^2 \gamma \)**:
To find \( \cos^2 \gamma \), we need to simplify the equation:
\[ \frac{1}{4} + \frac{2}{4} + \cos^2 \gamma = 1 \quad \Rightarrow \quad \frac{3}{4} + \cos^2 \gamma = 1 \]
\[ \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4} \]

7. **Calculating \( \cos \gamma \)**:
Taking the square root gives:
\[ \cos \gamma = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

8. **Finding the Angle \( \gamma \)**:
The angle \( \gamma \) can be found using the cosine value:
\[ \gamma = \cos^{-1} \left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Final Answer:
The acute angle made by the line with the z-axis is \( \frac{\pi}{3} \) radians or \( 60^\circ \).