A line AB in three-dimensional space makes angles `45^(@) and 120^(@)` with the positive X-axis and The positive Y-axis, respectively. If AB makes an acute angle`theta` with the positive Z-axis, then `theta` equals

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the problem, we need to determine the acute angle \( \theta \) that the line AB makes with the positive Z-axis. We know that the line makes angles of \( 45^\circ \) with the X-axis and \( 120^\circ \) with the Y-axis.

### Step-by-step Solution:

1. **Understanding the Angles**:
- Let \( \alpha = 45^\circ \) (angle with the X-axis)
- Let \( \beta = 120^\circ \) (angle with the Y-axis)
- Let \( \gamma = \theta \) (angle with the Z-axis)

2. **Using the Cosine Relation**:
We use the relation that states:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Substituting the Known Values**:
Substitute \( \alpha \) and \( \beta \) into the equation:
\[ \cos^2(45^\circ) + \cos^2(120^\circ) + \cos^2(\theta) = 1 \]

4. **Calculating the Cosines**:
- \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \)
- \( \cos(120^\circ) = \cos(180^\circ - 60^\circ) = -\cos(60^\circ) = -\frac{1}{2} \)

5. **Substituting the Cosine Values**:
Now we substitute these values into the equation:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + \cos^2(\theta) = 1 \]
This simplifies to:
\[ \frac{1}{2} + \frac{1}{4} + \cos^2(\theta) = 1 \]

6. **Finding a Common Denominator**:
To combine the fractions, we can convert \( \frac{1}{2} \) to have a denominator of 4:
\[ \frac{2}{4} + \frac{1}{4} + \cos^2(\theta) = 1 \]

7. **Combining the Terms**:
Combine the fractions:
\[ \frac{3}{4} + \cos^2(\theta) = 1 \]

8. **Isolating \( \cos^2(\theta) \)**:
Rearranging gives:
\[ \cos^2(\theta) = 1 - \frac{3}{4} = \frac{1}{4} \]

9. **Taking the Square Root**:
Taking the square root of both sides:
\[ \cos(\theta) = \frac{1}{2} \]

10. **Finding \( \theta \)**:
The angle \( \theta \) that satisfies \( \cos(\theta) = \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

### Final Answer:
Thus, the acute angle \( \theta \) that the line AB makes with the positive Z-axis is:
\[ \theta = 60^\circ \]