If a line makes angles of measure `45^(@)` and `60^(@)` with the positive direction of the Y and Z axex respectively, then the angle made by the line with the positive directions of the X-axis is

Options

A. `30^(@)" or " 150^(@)`
B. `30^(@)" or " 120^(@)`
C. `60^(@)" or " 120^(@)`
D. `60^(@)" or " 150^(@)`

Answer

To find the angle made by the line with the positive direction of the X-axis, we can use the relationship between the angles made with the coordinate axes and their cosines. Let's denote:

- \( \alpha \) = angle made with the X-axis
- \( \beta \) = angle made with the Y-axis = \( 45^\circ \)
- \( \gamma \) = angle made with the Z-axis = \( 60^\circ \)

### Step 1: Use the relationship between the angles
We know that:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 2: Substitute the known values
Substituting \( \beta \) and \( \gamma \):
\[ \cos^2 \alpha + \cos^2(45^\circ) + \cos^2(60^\circ) = 1 \]

### Step 3: Calculate the cosine values
We know:
- \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \)
- \( \cos(60^\circ) = \frac{1}{2} \)

So, substituting these values:
\[ \cos^2 \alpha + \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 = 1 \]

### Step 4: Simplify the equation
Calculating the squares:
\[ \cos^2 \alpha + \frac{1}{2} + \frac{1}{4} = 1 \]

### Step 5: Combine the fractions
Combine \( \frac{1}{2} \) and \( \frac{1}{4} \):
\[ \frac{1}{2} = \frac{2}{4} \implies \cos^2 \alpha + \frac{2}{4} + \frac{1}{4} = 1 \]
\[ \cos^2 \alpha + \frac{3}{4} = 1 \]

### Step 6: Isolate \( \cos^2 \alpha \)
\[ \cos^2 \alpha = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 7: Take the square root
\[ \cos \alpha = \pm \frac{1}{2} \]

### Step 8: Determine the angle \( \alpha \)
The angles corresponding to \( \cos \alpha = \frac{1}{2} \) are:
- \( \alpha = 60^\circ \) (1st quadrant)
- \( \alpha = 120^\circ \) (2nd quadrant)

### Conclusion
Thus, the angle made by the line with the positive direction of the X-axis can be either \( 60^\circ \) or \( 120^\circ \).