A line makes an angle `(pi)/(4)` with each y-axis and z-axis. The angle that it makes with the x - axis is

Options

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

Answer

To find the angle that the line makes with the x-axis, we can use the relationship between the angles made with the coordinate axes. Let's denote the angles made with the x-axis, y-axis, and z-axis as α, β, and γ respectively.

Given:
- β = π/4 (angle with the y-axis)
- γ = π/4 (angle with the z-axis)

We need to find α (angle with the x-axis).

### Step-by-step Solution:

1. **Use the Cosine Relation:**
We know the relation for angles made with the coordinate axes:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

2. **Substitute Known Values:**
Substitute the known values of β and γ into the equation:
\[ \cos^2 \alpha + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \left(\frac{\pi}{4}\right) = 1 \]

3. **Calculate Cosine Values:**
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \quad \text{so} \quad \cos^2\left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Therefore, we can write:
\[ \cos^2 \alpha + \frac{1}{2} + \frac{1}{2} = 1 \]

4. **Simplify the Equation:**
Combine the terms:
\[ \cos^2 \alpha + 1 = 1 \]

5. **Isolate Cosine Term:**
Rearranging gives:
\[ \cos^2 \alpha = 0 \]

6. **Find α:**
Taking the square root of both sides:
\[ \cos \alpha = 0 \]
This implies:
\[ \alpha = \frac{\pi}{2} \quad \text{(or 90 degrees)} \]

### Final Answer:
The angle that the line makes with the x-axis is \( \frac{\pi}{2} \).