Trigonometric Equations Question 110
Question: The perimeter of $ \Delta ABC $ is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is
[IIT Screening 1992; DCE 1999]
Options:
A) $ \frac{\pi }{6} $
B) $ \frac{\pi }{3} $
C) $ \frac{\pi }{2} $
D) $ \pi $
Show Answer
Answer:
Correct Answer: A
Solution:
- We have $ a+b+c=\frac{6(\sin A+\sin B+\sin C)}{3} $
Þ $ k(\sin A+\sin B+\sin C)=2(\sin A+\sin B+\sin C) $ , where $ k=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} $
Þ $ k=2 $ , $ (\because \sin A+\sin B+\sin C\ne 0) $
$ \therefore $ $ \frac{a}{\sin A}=2 $
Þ $ \sin A=\frac{1}{2} $ , ( $ \because $ $ a=1 $ ) Þ $ A=\frac{\pi }{6} $ .
BETA
