A line makes angles `alpha, beta, gamma` with X, Y, Z axes respectively. If `alpha=beta` and `gamma=45^(@)`, then `alpha+beta+gamma=`

Options

A. `165^(@)`
B. `180^(@)`
C. `135^(@)`
D. `120^(@)`

Answer

To solve the problem step by step, we need to find the sum of the angles \(\alpha\), \(\beta\), and \(\gamma\) given the conditions that \(\alpha = \beta\) and \(\gamma = 45^\circ\).

### Step 1: Understanding the relationship between angles and direction cosines
The direction cosines of a line making angles \(\alpha\), \(\beta\), and \(\gamma\) with the X, Y, and Z axes respectively are given by:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 2: Substitute the known values
Since we know that \(\alpha = \beta\) and \(\gamma = 45^\circ\), we can substitute these values into the equation:
\[ \cos^2 \alpha + \cos^2 \alpha + \cos^2 45^\circ = 1 \]
This simplifies to:
\[ 2\cos^2 \alpha + \left(\frac{1}{\sqrt{2}}\right)^2 = 1 \]

### Step 3: Calculate \(\cos^2 45^\circ\)
The value of \(\cos 45^\circ\) is \(\frac{1}{\sqrt{2}}\), so:
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 4: Substitute \(\cos^2 45^\circ\) into the equation
Now we can rewrite the equation:
\[ 2\cos^2 \alpha + \frac{1}{2} = 1 \]

### Step 5: Solve for \(\cos^2 \alpha\)
Subtract \(\frac{1}{2}\) from both sides:
\[ 2\cos^2 \alpha = 1 - \frac{1}{2} \]
\[ 2\cos^2 \alpha = \frac{1}{2} \]
Now divide by 2:
\[ \cos^2 \alpha = \frac{1}{4} \]

### Step 6: Find \(\cos \alpha\)
Taking the square root gives:
\[ \cos \alpha = \frac{1}{2} \]
This implies:
\[ \alpha = 60^\circ \quad \text{(since angles are positive)} \]

### Step 7: Find \(\beta\)
Since \(\alpha = \beta\), we have:
\[ \beta = 60^\circ \]

### Step 8: Calculate \(\alpha + \beta + \gamma\)
Now we can find the sum:
\[ \alpha + \beta + \gamma = 60^\circ + 60^\circ + 45^\circ = 165^\circ \]

### Final Answer
Thus, the value of \(\alpha + \beta + \gamma\) is:
\[ \boxed{165^\circ} \]