In any triangle ABC, find the value of a (B-C)+b (C-A)+c (A-B). - Vidyadip

Question

In any triangle ABC, find the value of $\text{a}\sin(\text{B}-\text{C})+\text{b}\sin(\text{C}-\text{A})+\text{c}\sin(\text{A}-\text{B}).$

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Answer

$\text{a}\sin(\text{B}-\text{C})+\text{b}\sin(\text{C}-\text{A})+\text{c}\sin(\text{A}-\text{B})$ $=\text{k}\sin\text{A}\sin(\text{B}-\text{C})+\text{k}\sin\text{B}\sin(\text{C}-\text{A})+\text{k}\sin\text{C}\sin(\text{A}-\text{B})$ $=\text{k}\sin(\pi-(\text{B + C}))\sin(\text{B}-\text{C})+\text{k}\sin(\pi-(\text{C + A}))\sin(\text{C}-\text{A})\\+\text{k}\sin(\pi-(\text{A + B}))\sin(\text{A}-\text{B})$ $=\text{k}\sin(\text{B + C})\sin(\text{B}-\text{C})+\text{k}\sin(\text{C + A})\\\sin(\text{C}-\text{A})+\text{k}\sin(\text{A + B})\sin(\text{A}-\text{B})$ $=\text{k}\big[\sin^2\text{B}-\sin^2\text{C}+\sin^2\text{C}-\sin^2\text{A}+\sin^2\text{A}-\sin^2\text{B}\big]$ $=\text{k}\times0 = 0$

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