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Question

If a,b,c denote the lengths of the sides of a triangle opposite angles A,B,C of a triangle ABC, then the correct relation among a,b,c,A,B and C is given by

A
(b+c)sin(B+C2)=acos(A2)
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B
(bc)cos(A2)=asin(BC2)
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C
(bc)sin(BC2)=acos(A2)
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D
(bc)cos(A2)=2asin(BC2)
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Solution

The correct option is B (bc)cos(A2)=asin(BC2)
Take b=ksinB;c=ksinC;a=ksinA
bca=k(sinBsinC)ksinA
Using tranformation angle formulae, we get
k(sinBsinC)ksinA=2cos(B+C2)sin(BC2)2sin(A2)cos(A2)
We know that A+B+C=1800
B+C2=900A2
cos(B+C2)=cos(900A2)=sin(A2)
Substituting cos(B+C2)=sin(A2) we get
bca=2sinA2sin(BC2)2sinA2cosA2
=sin(BC2)cosA2
(bc)cosA2=sin(BC2)

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