Trigonometric Identities Question 222
Question: If $ \alpha ,,\beta ,,\gamma \in ,( 0,,\frac{\pi }{2} ) $ , then $ \frac{\sin ,(\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma } $ is
Options:
A) < 1
B) >1
C) = 1
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
We have $ \sin \alpha +\sin \beta +\sin \gamma -\sin (\alpha +\beta +\gamma ) $ $ =\sin \alpha +\sin \beta +\sin \gamma -\sin \alpha \cos \beta \cos \gamma $ $ -\cos \alpha \sin \beta \cos \gamma -\cos \alpha \cos \beta \sin \gamma +\sin \alpha \sin \beta \sin \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 >0 $
$ \therefore \sin \alpha +\sin \beta +\sin \gamma >\sin (\alpha +\beta +\gamma ) $
$ \Rightarrow \frac{\sin (\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }<1 $ .
BETA
