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PROVE THAT acosA+bcosB+ccosC=4RsinAsinBsinC plz give me the full solution as quck as possible

PROVE THAT
acosA+bcosB+ccosC=4RsinAsinBsinC
 
plz give me the full solution as quck as possible

Grade:11

1 Answers

Vikas TU
14149 Points
4 years ago
Dear student 
Area of traingle = 1/2 bc SinA = 1/2 ac SinB  = 1/2 abbSinnC 
R = (a*b*c)/4*Area ABC 
4R SinA SinB SinC = 4* abc/4A  * SinA *SinB *SinC 
4R SinA SinB SinC = 2a *SinB *SinC 
Use formula a/sin A = b/SinB = c/SinC  = k 
put a= ksinA 
and use 2sinA sinB = Cos(A-B)-Cos(A+B)
Put CosC = Cos (pi-(A+B))
It comes out to be 
K/2 (Sin2A +Sin2B ) +Sin2C 
k/2 (2SinA*CosA + 2SinB*CosB + 2SinC*CosC ) 
KsinsinA cosA + KsinBCosB + KSinC CosC 
= aCosA + bCosB + cCosC 
Hope this help 
Good Luck 
 

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