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in any triangle, show that sin(A + B + C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB, where A, B, C : are angles.

in any triangle, show that sin(A + B + C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB, where A, B, C : are angles.

Grade:11

1 Answers

Arun
25758 Points
5 years ago
Dear Student
 
  Use the formula 
sin(a + b) = sina cosb + sinb cos a 

Then sin(a + b + c) 
= sin((a + b) + c) 
= sin(a + b) cosc + cos(a + b) sin c 

Then use the formula 
cos(a + b) = cosa cosb - sina sinb to get 

sin(a + b) cosc + cos(a + b) sin c 
= (sina cosb + sinb cosa) cosc + (cosa cosb - sina sinb) sinc 
= sina cosb cosc + sinb cosa cosc + sinc cosa cosb - sina sinb sinc
 
Regards
Arun (askIITians forum expert)

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