A line makes an angle of `60^0` with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

Options

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

Explanation

To find the acute angle made by the line with the Z-axis, we can use the concept of direction cosines. Here’s a step-by-step solution:

### Step 1: Understand the problem
The line makes an angle of \(60^\circ\) with both the X-axis and Y-axis. We need to find the angle it makes with the Z-axis.

### Step 2: Define the angles
Let:
- \( \alpha \) be the angle with the X-axis,
- \( \beta \) be the angle with the Y-axis,
- \( \gamma \) be the angle with the Z-axis.

From the problem, we have:
- \( \alpha = 60^\circ \)
- \( \beta = 60^\circ \)

### Step 3: Use the direction cosine identity
The direction cosines satisfy the equation:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Substituting the known values:
\[ \cos^2 60^\circ + \cos^2 60^\circ + \cos^2 \gamma = 1 \]

### Step 4: Calculate the cosines
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]
Thus:
\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Now substituting this into the equation:
\[ \frac{1}{4} + \frac{1}{4} + \cos^2 \gamma = 1 \]

### Step 5: Simplify the equation
Combine the terms:
\[ \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]
So we have:
\[ \frac{1}{2} + \cos^2 \gamma = 1 \]

### Step 6: Solve for \( \cos^2 \gamma \)
Subtract \( \frac{1}{2} \) from both sides:
\[ \cos^2 \gamma = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 7: Find \( \cos \gamma \)
Taking the square root:
\[ \cos \gamma = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 8: Determine \( \gamma \)
Now, we find the angle \( \gamma \):
\[ \gamma = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45^\circ \]

### Final Answer
The acute angle made by the line with the Z-axis is \( 45^\circ \).

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