If a line makes angles `alpha, beta, gamma` when the positive direction of coordinates axes, then write the value of `sin^(2) alpha + sin^(2) beta + sin^(2) gamma`.

Options

A. 3
B. 2
C. 1
D. 4

Answer

To solve the problem, we need to find the value of \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma \) given that a line makes angles \( \alpha, \beta, \gamma \) with the positive direction of the coordinate axes.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \( \alpha, \beta, \gamma \) with the positive x, y, and z axes are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \]

2. **Using the Property of Direction Cosines**:
We know that the sum of the squares of the direction cosines equals 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Using the Pythagorean Identity**:
From the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
We can express \( \cos^2 \alpha, \cos^2 \beta, \cos^2 \gamma \) in terms of \( \sin^2 \alpha, \sin^2 \beta, \sin^2 \gamma \):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]
\[ \cos^2 \beta = 1 - \sin^2 \beta \]
\[ \cos^2 \gamma = 1 - \sin^2 \gamma \]

4. **Substituting into the Equation**:
Substitute these expressions into the equation for the sum of squares of direction cosines:
\[ (1 - \sin^2 \alpha) + (1 - \sin^2 \beta) + (1 - \sin^2 \gamma) = 1 \]

5. **Simplifying the Equation**:
This simplifies to:
\[ 3 - (\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma) = 1 \]

6. **Isolating the Sine Squares**:
Rearranging gives:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 3 - 1 \]
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]

### Final Result:
Thus, the value of \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma \) is:
\[ \boxed{2} \]