proove that:acosA + bcosB + ccosC = 8(area of triangle)2/(abc)
proove that:
acosA + bcosB + ccosC = 8(area of triangle)2/(abc)
rahul agrawal , 7 Years ago
Grade 11
Trigonometry> proove that: acosA + bcosB + ccosC = 8(ar...
1 Answers
Arun
Last Activity: 7 Years ago
Dear Rahul
Then ... bc sin A = ca sin B = ab sin C = 2Δ ... (1)
Also ... bc sin A = 2Δ
. . . . . sin A = 2Δ / bc.
Similarly, ..... sin B = 2Δ / ca ... and ... sin C = 2Δ / ab. ... (2)
.........................................
... LHS
. ▬▬▬
= a cos A + b cos B + c cos C, ... where ... a/sin A = ... = k
= (k/2) [ 2 sin A cos A + 2 sin B cos B + 2 sin C cos C ]
= (k/2) [ ( sin 2A + sin 2B ) + sin 2C ]
= (k/2) [ 2 sin (A+B)· cos(A-B) + sin 2C ]
= (k/2) [ 2 sin C. cos(A-B) + 2 sin C cos C ]
= k sin C [ cos(A-B) + cos C ] ... here .. cos C = cos [ π - (A+B) ] = - cos (A+B)
= k sin C [ cos(A-B) - cos(A+B) ]
= k sin C [ 2 sin A sin B ]
= 2 ( k sin A ) sin B sin C
= 2 a sin B sin C
= Middle Side
▬▬▬▬▬▬▬
= 2a [ 2Δ / ca ] [ 2Δ / ab ] ...... from (2)
= 8 Δ² / ( abc )
= RHS
Regards
Arun (askIITians forum expert)
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