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Vector Algebra

Question
CBSEENMA12034273

If A, B and C are the vertices of a triangle ABC, prove Sine Formula that
fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction.

Solution

 Let space BC with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space CA with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space AB with rightwards arrow on top space equals space straight c with rightwards arrow on top space so space that space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space minus straight c with rightwards arrow on top
therefore space space space space straight a with rightwards arrow on top space cross times space straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space minus straight a with rightwards arrow on top cross times space straight c with rightwards arrow on top
therefore space space space space 0 with rightwards arrow on top space plus space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top
therefore space space space space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top
therefore space space space space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top close vertical bar
therefore space space straight a space straight b space sin space left parenthesis 180 degree minus straight C right parenthesis space equals space straight c space straight a space sin space left parenthesis 180 degree space minus space straight B right parenthesis
therefore space space space straight a space straight b space sin space straight C space equals space straight c space straight a space sin space straight B

Dividing both sides by a, b,  c we get,
               fraction numerator sin space straight C over denominator straight c end fraction space equals space fraction numerator sin space straight B over denominator straight b end fraction comma space space space space space space space therefore space space space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction
Similarly,          fraction numerator straight c over denominator sin space straight C end fraction space equals space fraction numerator straight a over denominator sin space straight A end fraction                                                 ...(1)
From (1) and (2), we have
             fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator straight b over denominator sin space straight B end fraction space equals space fraction numerator straight c over denominator sin space straight C end fraction.