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If the angles A, B and C of a triangle a...

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression `(a)/(c) sin 2C + (c)/(a) sin 2A` is

A

`(1)/(2)`

B

`(sqrt(3))/(2)`

C

1

D

`sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
D
Since A,B,C are in AP
`rArr 2B=A+C i.e., angle B =60^(@)`
`:. (a)/(c)(2 sin C cos C)+(c)/(a)(2 sin A cos A)`
`=2k(a cos C+c cos A)`
[ Using `(a)/(ainA)=(b)/(sinB)=(c)/(sinC)=(c)/(sin C)=(1)/(k)]`
`=2k(b)`
`=2 sin B " " [ "using" b=a cos C+c cos A]`
`=sqrt(3)`
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