If a line makes angles of `30^(@) and 45^(@)` with X-axis and Y-axis respectively, then the angle made by it with Z-axis is

Options

A. `45^(@)`
B. `60^(@)`
C. `120^(@)`
D. does not exist

Answer

To solve the problem, we need to find the angle made by a line with the Z-axis, given that it makes angles of \(30^\circ\) and \(45^\circ\) with the X-axis and Y-axis, respectively.

### Step-by-Step Solution:

1. **Understand Direction Cosines**:
The direction cosines of a line are defined as:
- \( l = \cos(\alpha) \) (angle with X-axis)
- \( m = \cos(\beta) \) (angle with Y-axis)
- \( n = \cos(\gamma) \) (angle with Z-axis)

Here, \(\alpha = 30^\circ\) and \(\beta = 45^\circ\).

2. **Calculate \( l \) and \( m \)**:
- For \( \alpha = 30^\circ \):
\[ l = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
- For \( \beta = 45^\circ \):
\[ m = \cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

3. **Use the Property of Direction Cosines**:
The sum of the squares of the direction cosines is equal to 1:
\[ l^2 + m^2 + n^2 = 1 \]

4. **Substitute the Values**:
Substitute \( l \) and \( m \) into the equation:
\[ \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + n^2 = 1 \]
This simplifies to:
\[ \frac{3}{4} + \frac{2}{4} + n^2 = 1 \]

5. **Combine the Terms**:
Combine the fractions:
\[ \frac{3}{4} + \frac{2}{4} = \frac{5}{4} \]
Therefore, the equation becomes:
\[ \frac{5}{4} + n^2 = 1 \]

6. **Solve for \( n^2 \)**:
Rearranging gives:
\[ n^2 = 1 - \frac{5}{4} = -\frac{1}{4} \]

7. **Analyze the Result**:
Since \( n^2 \) cannot be negative (as squares of real numbers are non-negative), we conclude that there is no real value for \( n \).

### Final Conclusion:
The angle made by the line with the Z-axis does not exist in real numbers.