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Math Help - Search for an 'elegant' solution

  1. #1
    Member great_math's Avatar
    Joined
    Oct 2008
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    132

    Search for an 'elegant' solution

    Given,

    x=\frac{\sin^3 p}{\cos^2 p  }

    y=\frac{\cos^3 p}{\sin^2 p  }

    and \sin p+\cos p=\frac1{2}

    Find the value of x+y

    here is my solution:

    Spoiler:

    2\sin p\cos p=\frac{-3}{4}

    \Rightarrow \sin p\cos p=\frac{-3}{8}

    x+y =\frac{\sin^5+\cos^5}{\sin^2p\cos^2p}

    \Rightarrow x+y=\frac{(\sin p+\cos p)(\sin^4p-\sin^3p\cos p+\sin^2p\cos^2p-\sin p\cos^3p+\cos^4p)}{\sin^2p\cos^2p}

    \Rightarrow x+y=\frac{(\sin p+\cos p)^4-5\sin^3p\cos p-5\sin^2p\cos^2p-5\sin p\cos^3p}{2\sin^2p\cos^2p}

    \Rightarrow x+y=\frac{\frac1{16}-5\sin p\cos p(\sin^2p-\sin p\cos p+\cos^2p)}{\frac{2\times9}{64}}

    \Rightarrow x+y=\frac{\frac1{16}-5\sin p\cos p\{(\sin p+\cos p)^2-\sin p\cos p\}}{\frac{2\times9}{64}}

    \Rightarrow x+y=\frac{\frac1{16}+\frac{5\times3}{8}(\frac1{4}+  \frac{3}{8})}{\frac{2\times9}{64}}

    \Rightarrow x+y=\frac{\frac{79}{64}}{\frac{18}{64}}


    \Rightarrow x+y=\frac{79}{18}
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  2. #2
    Super Member malaygoel's Avatar
    Joined
    May 2006
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    India
    Posts
    648
    Quote Originally Posted by great_math View Post
    Given,

    x=\frac{\sin^3 p}{\cos^2 p  }

    y=\frac{\cos^3 p}{\sin^2 p  }

    and \sin p+\cos p=\frac1{2}

    Find the value of x+y

    here is my solution:


    2\sin p\cos p=\frac{-3}{4}

    I hope you find this elegant!!!


    x=\frac{sin^3p}{cos^2p}=\frac{sinp}{cos^2p}-sinp

    simlilarly,

    y=\frac{cosp}{sin^2p}-cosp

    x+y

    =\frac{sinp}{cos^2p}-sinp + \frac{cosp}{sin^2p}-cosp

    =\frac{sinp}{cos^2p}+ \frac{cosp}{sin^2p}-cosp-sinp

    = \frac{cos^3p+sin^3p}{cos^2psin^2p}-(cosp+sinp)

    = \frac{(cosp+sinp)^3-3cospsinp(cosp+sinp)}{(cospsinp)^2}-(cosp+sinp)

    = \frac{\frac{1}{8}+\frac{9}{16}}{\frac{9}{64}}-\frac{1}{2}


    =\frac{79}{18}
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  3. #3
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288
    I imagine that the use of a calculator introduces an element of inelegance into the solution, otherwise we could simply state that

    2sinpcosp=sin2p=\frac{-3}{4}

    and find the value of p and use that to solve the equation
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