Principle Of Mathematical Induction

Question
CBSEENMA11014341

Deduce the sine formula from cosine formula.

Solution

We have,
           fraction numerator straight a squared over denominator sin squared straight A end fraction equals fraction numerator straight a squared over denominator 1 minus cos squared straight A end fraction equals fraction numerator straight a squared over denominator 1 minus open square brackets begin display style fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction end style close square brackets squared end fraction equals fraction numerator straight a squared over denominator begin display style fraction numerator 4 straight b squared straight c squared minus left parenthesis straight b squared plus straight c squared minus straight a squared right parenthesis squared over denominator 4 straight b squared straight c squared end fraction end style end fraction
                      equals fraction numerator 4 straight a squared straight b squared straight c squared over denominator left parenthesis 2 bc plus straight b squared plus straight c squared minus straight a squared right parenthesis space left parenthesis 2 bc minus straight b squared minus straight c squared plus straight a squared right parenthesis end fraction
                      equals fraction numerator 4 straight a squared straight b squared straight c squared over denominator left square bracket left parenthesis straight b plus straight c right parenthesis squared minus straight a squared right square bracket space left square bracket straight a squared minus left parenthesis straight b minus straight c right parenthesis squared right square bracket end fraction
                       equals space fraction numerator 4 straight a squared straight b squared straight c squared over denominator left parenthesis straight b plus straight c plus straight a right parenthesis left parenthesis straight b plus straight c minus straight a right parenthesis left parenthesis straight a plus straight b minus straight c right parenthesis left parenthesis straight a minus straight b plus straight c right parenthesis end fraction
           fraction numerator straight a over denominator sin space straight A end fraction space equals space fraction numerator 2 abc over denominator square root of 2 straight s left parenthesis 2 straight s minus 2 straight a right parenthesis space left parenthesis 2 straight s minus 2 straight c right parenthesis space left parenthesis 2 straight s minus 2 straight b right parenthesis end root end fraction
                      equals space fraction numerator abc over denominator 2 square root of straight s left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis left parenthesis straight s minus straight c right parenthesis end root end fraction  [∵ sin A is positive as straight A less than 180 degree]
Similarly, we can show
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Hence, straight a over sinA equals straight b over sinB equals straight c over sinC
Note:  Area of the traingle ABC is denoted by A.
                       

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Some More Questions From Principle of Mathematical Induction Chapter

Find 1 radian angle in degrees, minutes, seconds.