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

Options

A. `(pi)/2`
B. `(pi)/3`
C. `(pi)/4`
D. `(5pi)/12`

Explanation

To find the angle made by the line with the z-axis, we can use the relationship between the angles made with the x-axis, y-axis, and z-axis. The angles made with the axes are denoted as \( \alpha \), \( \beta \), and \( \gamma \) for the x-axis, y-axis, and z-axis respectively.

Given:
- \( \alpha = \frac{\pi}{3} \) (angle with x-axis)
- \( \beta = \frac{\pi}{4} \) (angle with y-axis)

We need to find \( \gamma \) (angle with z-axis).

### Step 1: Use the relationship between angles
The relationship between the angles made with the axes is given by:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 2: Calculate \( \cos \alpha \) and \( \cos \beta \)
1. Calculate \( \cos \alpha \):
\[ \cos \alpha = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

2. Calculate \( \cos \beta \):
\[ \cos \beta = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

### Step 3: Substitute values into the equation
Substituting the values of \( \cos \alpha \) and \( \cos \beta \) into the equation:
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \cos^2 \gamma = 1 \]
\[ \frac{1}{4} + \frac{2}{4} + \cos^2 \gamma = 1 \]
\[ \frac{3}{4} + \cos^2 \gamma = 1 \]

### Step 4: Solve for \( \cos^2 \gamma \)
Rearranging the equation gives:
\[ \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Find \( \gamma \)
Taking the square root:
\[ \cos \gamma = \sqrt{\frac{1}{4}} = \frac{1}{2} \]
Thus, the angle \( \gamma \) is:
\[ \gamma = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Final Answer
The angle made by the line with the z-axis is \( \frac{\pi}{3} \).

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