Question:
The minimum value of the expression \(\sin \alpha + \sin \beta + \sin \gamma\) where \(\alpha, \beta, \gamma\) are real numbers satisfying \(\alpha + \beta + \gamma = \pi\) is
The minimum value of the expression \(\sin \alpha + \sin \beta + \sin \gamma\) where \(\alpha, \beta, \gamma\) are real numbers satisfying \(\alpha + \beta + \gamma = \pi\) is
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Test extreme values; sine function can be negative.
Updated On: Apr 23, 2026
- positive
- zero
- negative
- -3
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The Correct Option is C
Solution and Explanation
Step 1: Formula / Definition}
\[ \sin \alpha + \sin \beta + \sin \gamma \]
Step 2: Calculation / Simplification}
For \(\alpha = \beta = -\frac{\pi}{2}, \gamma = 2\pi\):
\(\alpha+\beta+\gamma = -\frac{\pi}{2} - \frac{\pi}{2} + 2\pi = \pi\)
\(\sin \alpha + \sin \beta + \sin \gamma = -1 - 1 + 0 = -2\)
Since \(-2<0\), minimum is negative.
Step 3: Final Answer
\[ \text{negative} \]
\[ \sin \alpha + \sin \beta + \sin \gamma \]
Step 2: Calculation / Simplification}
For \(\alpha = \beta = -\frac{\pi}{2}, \gamma = 2\pi\):
\(\alpha+\beta+\gamma = -\frac{\pi}{2} - \frac{\pi}{2} + 2\pi = \pi\)
\(\sin \alpha + \sin \beta + \sin \gamma = -1 - 1 + 0 = -2\)
Since \(-2<0\), minimum is negative.
Step 3: Final Answer
\[ \text{negative} \]
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