If `alpha,beta` and `gamma` are angles of a triangle then `sinalpha +sinbeta+singamma` is equal to -

Options

A. `2"cos"alpha/2"cos"beta/2"cos"gamma/2`
B. `8cosalpha cosbetacosgamma`
C. `cos((alpha+beta)/2)cos((beta+gamma)/2)`
D. `4"cos"alpha/2 "cos"beta/2 "cos"gamma/2`

Answer

To solve the problem, we need to find the value of \( \sin \alpha + \sin \beta + \sin \gamma \) where \( \alpha, \beta, \) and \( \gamma \) are the angles of a triangle.

### Step-by-Step Solution:

1. **Understanding the Angles of a Triangle**:
- The sum of the angles in any triangle is \( 180^\circ \).
- Therefore, we can write:
\[ \alpha + \beta + \gamma = 180^\circ \]

2. **Using the Sine Function**:
- We need to find \( \sin \alpha + \sin \beta + \sin \gamma \).
- We can use the sine addition formula or properties of sine to express one of the angles in terms of the others.

3. **Expressing One Angle**:
- Let's express \( \gamma \) in terms of \( \alpha \) and \( \beta \):
\[ \gamma = 180^\circ - \alpha - \beta \]

4. **Applying the Sine Function**:
- Now, we can substitute \( \gamma \) into the sine function:
\[ \sin \gamma = \sin(180^\circ - \alpha - \beta) = \sin(\alpha + \beta) \]
- Using the sine addition formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

5. **Combining the Sine Values**:
- Now we can write:
\[ \sin \alpha + \sin \beta + \sin \gamma = \sin \alpha + \sin \beta + \sin(\alpha + \beta) \]
- Substituting for \( \sin(\alpha + \beta) \):
\[ = \sin \alpha + \sin \beta + \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

6. **Simplifying the Expression**:
- Combine like terms:
\[ = \sin \alpha (1 + \cos \beta) + \sin \beta (1 + \cos \alpha) \]

7. **Conclusion**:
- The expression \( \sin \alpha + \sin \beta + \sin \gamma \) does not simplify to a constant value but depends on the specific angles \( \alpha, \beta, \) and \( \gamma \).
- However, for specific cases (like \( \alpha = 90^\circ, \beta = 90^\circ, \gamma = 0^\circ \)), we can evaluate it to find specific values.

### Final Answer:
Thus, \( \sin \alpha + \sin \beta + \sin \gamma \) does not have a fixed value but can be evaluated based on the angles of the triangle.