Principle Of Mathematical Induction

Question
CBSEENMA11014334

Deduce projection formula from cosine formula.

Solution

In any triangle, we have to show that
                   a = b cosC + c cosB
                   b = c cosA + a cosC
                   c = a cosB + b cosA
R.H.S. = straight b space cosC space plus space straight c space cosB space equals space straight b open square brackets fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction close square brackets plus straight c open square brackets fraction numerator straight a squared plus straight c squared minus straight b squared over denominator 2 ac end fraction close square brackets
          = fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 straight a end fraction plus fraction numerator straight a squared plus straight c squared minus straight b squared over denominator 2 straight a end fraction space equals space fraction numerator straight a squared plus straight b squared minus straight c squared plus straight a squared plus straight c squared minus straight b squared over denominator 2 straight a end fraction
          = fraction numerator 2 straight a squared over denominator 2 straight a end fraction equals straight a equals straight L. straight H. straight S.
Similarly, we can show that
                   b = c cosA + a cosC
                   c = a cosB + b cosA
                  

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