The minimum value of the expression `sin alpha + sin beta + sin gamma,` where `alpha, beta,gamma` are the real numbers satisfying `alpha + beta + gamma= pi`, is

Options

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

Answer

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

### Step 1: Use the constraint
Given the constraint \( \alpha + \beta + \gamma = \pi \), we can express \( \gamma \) in terms of \( \alpha \) and \( \beta \):
\[ \gamma = \pi - \alpha - \beta \]

### 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 property of sine that \( \sin(\pi - x) = \sin x \), we have:
\[ \sin \alpha + \sin \beta + \sin(\alpha + \beta) \]

### Step 3: Apply the sine addition formula
Using the sine addition formula:
\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Thus, the expression becomes:
\[ \sin \alpha + \sin \beta + \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

### Step 4: Analyze the expression
To find the minimum value, we can analyze the behavior of the sine function. The sine function achieves its minimum value of 0 when its argument is 0 or an integer multiple of \( \pi \). Since \( \alpha, \beta, \gamma \) are angles that sum to \( \pi \), we can see that the minimum occurs when two angles are \( 0 \) and the third angle is \( \pi \).

### Step 5: Conclusion
Thus, the minimum value of \( \sin \alpha + \sin \beta + \sin \gamma \) occurs when one of the angles is \( 0 \) and the others are \( \pi \) or \( 0 \). Therefore, the minimum value of the expression is:
\[ \sin 0 + \sin 0 + \sin \pi = 0 + 0 + 0 = 0 \]

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

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