If `alpha, beta, gamma` are direction angles of a line and `alpha = 60^(@), beta = 45^(@)`, then `gamma` =_________

Options

A. `30^(@) or 90^(@)`
B. `45^(@) or 60^(@)`
C. `90^(@) or 30^(@)`
D. `60^(@) or 120^(@)`

Answer

To find the value of \( \gamma \) given that \( \alpha = 60^\circ \) and \( \beta = 45^\circ \), we can use the property of direction angles, which states:

\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step-by-step Solution:

1. **Substitute the values of \( \alpha \) and \( \beta \)**:
\[ \cos^2(60^\circ) + \cos^2(45^\circ) + \cos^2(\gamma) = 1 \]

2. **Calculate \( \cos(60^\circ) \) and \( \cos(45^\circ) \)**:
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)

3. **Square the cosine values**:
\[ \cos^2(60^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \cos^2(45^\circ) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} \]

4. **Substitute these squared values back into the equation**:
\[ \frac{1}{4} + \frac{1}{2} + \cos^2(\gamma) = 1 \]

5. **Convert \( \frac{1}{2} \) to a fraction with a common denominator**:
\[ \frac{1}{2} = \frac{2}{4} \]
So,
\[ \frac{1}{4} + \frac{2}{4} + \cos^2(\gamma) = 1 \]

6. **Combine the fractions**:
\[ \frac{3}{4} + \cos^2(\gamma) = 1 \]

7. **Isolate \( \cos^2(\gamma) \)**:
\[ \cos^2(\gamma) = 1 - \frac{3}{4} = \frac{1}{4} \]

8. **Take the square root to find \( \cos(\gamma) \)**:
\[ \cos(\gamma) = \pm \frac{1}{2} \]

9. **Determine the angles corresponding to \( \cos(\gamma) \)**:
- \( \cos(\gamma) = \frac{1}{2} \) gives \( \gamma = 60^\circ \)
- \( \cos(\gamma) = -\frac{1}{2} \) gives \( \gamma = 120^\circ \)

### Final Answer:
Thus, \( \gamma \) can be either \( 60^\circ \) or \( 120^\circ \).

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