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Maths-

General

Easy

Question

In $straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals$

abc
$2 a b c$
abc
0

hint Hint:

In this question we will use sine and cosine formula to find the required sum.

The correct answer is: $2 a b c$

$In space straight capital delta A B C comma sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis L e t space fraction numerator sin space A thin space over denominator a end fraction equals fraction numerator sin space B over denominator b end fraction equals fraction numerator sin space C over denominator c end fraction equals k cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction comma cos space A equals fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction comma cos space A equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction space space space space space space straight a cubed sin left parenthesis straight B minus straight C right parenthesis equals a cubed open parentheses fraction numerator k b left parenthesis a squared plus b squared minus c squared right parenthesis over denominator 2 a b end fraction minus fraction numerator k c left parenthesis c squared plus a squared minus b squared right parenthesis over denominator 2 a c end fraction close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses a squared plus b squared minus c squared minus c squared minus a squared plus b squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction open parentheses 2 b squared minus 2 c squared close parentheses equals fraction numerator k a squared over denominator 2 end fraction 2. open parentheses b squared minus c squared close parentheses equals k a squared open parentheses b squared minus c squared close parentheses s i m i l a r l y comma space space b cubed sin left parenthesis C minus A right parenthesis equals k b squared left parenthesis c squared minus a squared right parenthesis s i m i l a r l y comma space c cubed sin space left parenthesis A minus B right parenthesis equals k c squared left parenthesis a squared minus b squared right parenthesis N o w comma space sum a cubed sin invisible function application left parenthesis B minus C right parenthesis equals space straight a cubed sin left parenthesis straight B minus straight C right parenthesis plus straight b cubed sin left parenthesis straight C minus straight A right parenthesis plus straight c cubed left parenthesis straight A minus straight B right parenthesis equals k a squared open parentheses b squared minus c squared close parentheses plus k b squared left parenthesis c squared minus a squared right parenthesis plus k c squared left parenthesis a squared minus b squared right parenthesis equals k a squared b squared minus k a squared c squared plus k b squared c squared minus k a squared b squared plus k a squared c squared minus k b squared c squared equals 0.$

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