`alpha, beta, gamma` are real number satisfying `alpha+beta+gamma=pi`. The minimum value of the given expression `sin alpha+sin beta+sin gamma` is

Options

A. zero
B. `-3`
C. positive
D. negative

Explanation

To find the minimum value of the expression \( \sin \alpha + \sin \beta + \sin \gamma \) given that \( \alpha + \beta + \gamma = \pi \), we can follow these steps:

### Step 1: Use the property of sine
We know that the sine function is periodic and has a maximum value of 1 and a minimum value of -1. However, we will analyze the expression under the constraint that \( \alpha + \beta + \gamma = \pi \).

### Step 2: Rewrite the expression
Using the identity \( \gamma = \pi - (\alpha + \beta) \), we can rewrite the expression:
\[ \sin \alpha + \sin \beta + \sin \gamma = \sin \alpha + \sin \beta + \sin(\pi - (\alpha + \beta)) \]
Since \( \sin(\pi - x) = \sin x \), we have:
\[ \sin \gamma = \sin(\alpha + \beta) \]
Thus, the expression becomes:
\[ \sin \alpha + \sin \beta + \sin(\alpha + \beta) \]

### Step 3: Analyze the range of the sine function
The sine function is non-negative in the interval \( [0, \pi] \). Therefore, \( \sin \alpha \), \( \sin \beta \), and \( \sin(\alpha + \beta) \) are all non-negative as long as \( \alpha, \beta \) are in the range \( [0, \pi] \).

### Step 4: Find the minimum value
To find the minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \), we can consider the case when \( \alpha, \beta, \) and \( \gamma \) are all equal. If we set \( \alpha = \beta = \gamma = \frac{\pi}{3} \):
\[ \sin \alpha = \sin \beta = \sin \gamma = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin \alpha + \sin \beta + \sin \gamma = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]
This is a positive value.

### Step 5: Consider the case when one angle approaches 0
If we take \( \alpha = 0 \), \( \beta = 0 \), and \( \gamma = \pi \):
\[ \sin \alpha + \sin \beta + \sin \gamma = \sin(0) + \sin(0) + \sin(\pi) = 0 + 0 + 0 = 0 \]
However, since we are looking for the minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \) under the constraint that \( \alpha + \beta + \gamma = \pi \) and all angles are real, we can conclude that the expression can reach 0 but cannot go below it.

### Conclusion
Thus, the minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \) is \( 0 \).

### Final Answer
The minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \) is \( 0 \).

---