If `cos alpha + cos beta + cos gamma = 0`, where `0 lt alpha le (pi)/(2),0 lt beta le (pi)/(2),0 lt gamma le (pi)/(2)`. then what is the value of `sin alpha + sin beta + sin gamma`?

Options

A. 0
B. 3
C. `(5 sqrt2)/(2)`
D. `(3sqrt2)/(2)`

Answer

To solve the problem where \( \cos \alpha + \cos \beta + \cos \gamma = 0 \) and \( 0 < \alpha, \beta, \gamma \leq \frac{\pi}{2} \), we need to find the value of \( \sin \alpha + \sin \beta + \sin \gamma \).

### Step-by-Step Solution:

1. **Understanding the Range of Angles**:
- The angles \( \alpha, \beta, \gamma \) are all in the first quadrant, meaning \( 0 < \alpha, \beta, \gamma \leq \frac{\pi}{2} \). In this range, the cosine function is non-negative.

2. **Analyzing the Given Equation**:
- Given \( \cos \alpha + \cos \beta + \cos \gamma = 0 \), and since \( \cos \) is non-negative in the first quadrant, the only way for this sum to equal zero is if each cosine term is zero.
- Therefore, we conclude that:
\[ \cos \alpha = 0, \quad \cos \beta = 0, \quad \cos \gamma = 0 \]

3. **Finding the Angles**:
- The cosine of an angle is zero at \( \frac{\pi}{2} \). Thus, we have:
\[ \alpha = \frac{\pi}{2}, \quad \beta = \frac{\pi}{2}, \quad \gamma = \frac{\pi}{2} \]

4. **Calculating the Sine Values**:
- Now we calculate \( \sin \alpha, \sin \beta, \sin \gamma \):
\[ \sin \alpha = \sin \left(\frac{\pi}{2}\right) = 1 \]
\[ \sin \beta = \sin \left(\frac{\pi}{2}\right) = 1 \]
\[ \sin \gamma = \sin \left(\frac{\pi}{2}\right) = 1 \]

5. **Summing the Sine Values**:
- Therefore, we find:
\[ \sin \alpha + \sin \beta + \sin \gamma = 1 + 1 + 1 = 3 \]

### Final Answer:
The value of \( \sin \alpha + \sin \beta + \sin \gamma \) is \( 3 \).

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