Question

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

In any ∆ABC, find the value of $\sum _{}^{}a\left(\sin B-\sin C\right)$.

Answer 1

Using sine rule, we have

$$\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k \\ \Rightarrow a=k\sin A,b=k\sin B,c=k\sin C \end{gathered}$$

$$\begin{gathered} \therefore \sum _{}^{}a\left(\sin B-\sin C\right) \\ =\sum _{}^{}k\sin A\left(\sin B-\sin C\right) \end{gathered}$$

$=k\sum _{}^{}\sin A\left(\sin B-\sin C\right)$

$=k\left[\sin A\left(\sin B-\sin C\right)+\sin B\left(\sin C-\sin A\right)+\sin C\left(\sin A-\sin B\right)\right]$

$=k\left(\sin A\sin B-\sin A\sin C+\sin B\sin C-\sin B\sin A+\sin C\sin A-\sin C\sin B\right)$

$=k\times 0=0$