If a line OP through the origin O makes angles `alpha,45^(@)and60^(@)` with x, y and z axis respectively then the direction cosines of OP are.

Options

A. `1/sqrt(2),1/2,1/2`
B. `1/2,1/2,1/sqrt(2)`
C. `1/2,1/sqrt(2),1/2`
D. none of these

Answer

To find the direction cosines of the line OP that makes angles α, 45°, and 60° with the x-axis, y-axis, and z-axis respectively, we can follow these steps:

### Step 1: Understand Direction Cosines
The direction cosines of a line making angles α, β, and γ with the x, y, and z axes are given by:
- cos(α) for the x-axis
- cos(β) for the y-axis
- cos(γ) for the z-axis

In this case, we know:
- β = 45°
- γ = 60°

### Step 2: Calculate Known Direction Cosines
Now, we can calculate the direction cosines for the known angles:
- cos(β) = cos(45°) = √2/2
- cos(γ) = cos(60°) = 1/2

### Step 3: Use the Property of Direction Cosines
We know that the sum of the squares of the direction cosines is equal to 1:
\[ \cos^2(α) + \cos^2(β) + \cos^2(γ) = 1 \]

Substituting the known values:
\[ \cos^2(α) + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = 1 \]

### Step 4: Simplify the Equation
Calculating the squares:
- \(\left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}\)
- \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)

So the equation becomes:
\[ \cos^2(α) + \frac{1}{2} + \frac{1}{4} = 1 \]

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

Now, isolate cos²(α):
\[ \cos^2(α) = 1 - \frac{3}{4} \]
\[ \cos^2(α) = \frac{1}{4} \]

Taking the square root:
\[ \cos(α) = \frac{1}{2} \]

### Step 6: Write the Direction Cosines
Now we can write the direction cosines:
- cos(α) = 1/2
- cos(β) = √2/2
- cos(γ) = 1/2

Thus, the direction cosines of OP are:
\[ \left( \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2} \right) \]

### Final Answer
The direction cosines of the line OP are:
\[ \left( \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2} \right) \]

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