Question

If $ \alpha ,\beta ,\gamma \in \left[ 0,\frac{\pi }{2} \right], $ then the value of $ \frac{\sin (\alpha +\beta +\gamma )}{sin\text{ }\alpha +sin\,\beta +sin\text{ }\gamma } $ is

• A. $ <1 $
• B. $ =-1 $
• C. $ <0 $
• D. None of these

Answer 1

Answer: (A)

We have, $ sin\text{ }\alpha +sin\,\,\beta +sin\text{ }\gamma -sin\text{ }(\alpha +\beta +\gamma ) $ $ =sin\text{ }\alpha +sin\text{ }\beta +sin\text{ }\gamma -sin\alpha \text{ }cos\beta \text{ }cos\gamma $ $ -cos\text{ }\alpha \text{ }sin\text{ }\beta \text{ }cos\text{ }\gamma -cos\text{ }\alpha \text{ }cos\text{ }\beta \text{ }sin\text{ }\gamma $ $ +\text{ }sin\text{ }\alpha \text{ }sin\text{ }\beta \text{ }sin\text{ }\gamma $ $ =\sin \alpha (1-\cos \beta \cos \gamma )+\sin \beta (1-\cos \alpha \cos \gamma ) $ $ +\sin \gamma (1-\cos \alpha \cos \beta )+\sin \alpha \sin \beta \sin \gamma $ $ \therefore $ $ \sin \alpha +\sin \beta +\sin \gamma >\sin (\alpha +\beta +\gamma ) $ $ \Rightarrow $ $ \frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }<1 $