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    Maths-
    General
    Easy

    Question

    In triangle A B C comma sum left parenthesis a squared s i n invisible function application left parenthesis B minus C right parenthesis right parenthesis divided by left parenthesis s i n invisible function application A right parenthesis equals

    1. a squared b squared
    2. b squared c squared
    3. c squared a squared
    4. 0

    hintHint:

    In this question first we will us Sine Rule. Then we use the trigonometric formula sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses equals sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses to find the value of the sum

    The correct answer is: 0

      In this question we have to find the value of fraction numerator a squared sin open parentheses B minus C close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator b squared sin open parentheses C minus A close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator c squared sin open parentheses A minus B close parentheses over denominator sin open parentheses C close parentheses end fraction
      Step1: Using Sine Rule
      fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals k
      =>a equals space k sin open parentheses A close parentheses comma b equals k sin open parentheses B close parentheses comma space c equals k sin open parentheses C close parentheses
      By substituting these values in the given expression we get.
      fraction numerator k squared sin squared open parentheses A close parentheses sin open parentheses B minus C close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator k squared sin squared open parentheses B close parentheses sin open parentheses C minus A close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator k squared sin squared open parentheses C close parentheses sin open parentheses A minus B close parentheses over denominator sin open parentheses C close parentheses end fraction
      => k squared left parenthesis sin open parentheses A close parentheses sin open parentheses B minus C close parentheses plus sin open parentheses B close parentheses sin open parentheses C minus A close parentheses plus sin open parentheses C close parentheses sin open parentheses A minus B close parentheses right parenthesis
      we know that A plus B plus C equals straight pi
      =>A equals straight pi minus left parenthesis straight B plus straight C right parenthesis
      =>sin open parentheses A close parentheses equals sin open parentheses B plus C close parentheses
      So, we can write sin open parentheses A close parentheses equals sin open parentheses B plus C close parentheses comma sin open parentheses B close parentheses equals sin open parentheses A plus C close parentheses comma sin open parentheses C close parentheses equals sin open parentheses A plus B close parentheses
      by substituting these values in the above expression we get,
      =>k squared left parenthesis sin open parentheses B plus C close parentheses sin open parentheses B minus C close parentheses plus sin open parentheses C plus A close parentheses sin open parentheses C minus A close parentheses plus sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses right parenthesis
      Step2: Using trigonometric formula
      We know that sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses equals sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses
      => k squared left parenthesis sin squared open parentheses B close parentheses minus sin squared open parentheses C close parentheses plus sin squared open parentheses C close parentheses minus sin squared open parentheses A close parentheses plus sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses right parenthesis
      =>0
      Hence, the value of sum is 0.

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