`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 condition of the angles
Since we know that \( \alpha + \beta + \gamma = \pi \), we can express one of the angles in terms of the others. Let's express \( \gamma \) as:
\[ \gamma = \pi - \alpha - \beta \]

### Step 2: Substitute \( \gamma \) into the expression
Now, substitute \( \gamma \) into the expression:
\[ \sin \alpha + \sin \beta + \sin \gamma = \sin \alpha + \sin \beta + \sin(\pi - \alpha - \beta) \]
Using the sine identity \( \sin(\pi - x) = \sin x \), we have:
\[ \sin \gamma = \sin(\pi - \alpha - \beta) = \sin(\alpha + \beta) \]
Thus, the expression becomes:
\[ \sin \alpha + \sin \beta + \sin(\alpha + \beta) \]

### Step 3: Apply the sine addition formula
Using the sine addition formula, we can rewrite \( \sin(\alpha + \beta) \):
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
So, the expression now is:
\[ \sin \alpha + \sin \beta + \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

### Step 4: Analyze the expression
To find the minimum value, we can analyze the behavior of the sine function. The sine function varies between -1 and 1. Therefore, the sum \( \sin \alpha + \sin \beta + \sin(\alpha + \beta) \) will also have a minimum value.

### Step 5: Use the property of sine
From the properties of sine, we know that:
\[ \sin \alpha + \sin \beta + \sin \gamma \geq 0 \]
This is because the sine function is non-negative in the first and second quadrants, and since \( \alpha + \beta + \gamma = \pi \), at least one of the angles must be in the range where sine is non-negative.

### Conclusion
Thus, the minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \) is:
\[ \text{Minimum Value} = 0 \]