Question

Answer the following questions in one word or one sentence or as per exact requirement of the question. 

In any triangle ABC, find the value of \[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\ 

Answer 1

Using sine rule, we have 

$$a\sin (B-C)+b\sin (C-A)+c\sin (A-B)$$
$$=k\sin A\sin (B-C)+k\sin B\sin (C-A)+k\sin C\sin (A-B)$$
$$=k\sin [\pi -(B+C)]\sin (B-C)+k\sin [\pi -(C+A)]\sin (C-A)+k\sin [\pi -(A+B)]\sin (A-B)$$ 

$$=k[\sin (B+C)\sin (B-C)+\sin (C+A)\sin (C-A)+\sin (A+B)\sin (A-B)]$$
$$=k({\sin }^{2}B-{\sin }^{2}C+{\sin }^{2}C-{\sin }^{2}A+{\sin }^{2}A-{\sin }^{2}B)$$
$$=k\times 0$$
$$=0$$ 

Hence, the required value is 0.