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Math Help - trig prove

  1. #1
    Member srirahulan's Avatar
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    trig prove

    If sin^-1x+sin^-1y=\pi/3 Prove That x^2+xy+y^2=\frac{4}{3}
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  2. #2
    Member kalyanram's Avatar
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    Re: trig prove

    Quote Originally Posted by srirahulan View Post
    If sin^-1x+sin^-1y=\pi/3 Prove That x^2+xy+y^2=\frac{4}{3}
    The problem is incorrect it has to be x^2+xy+y^2=\frac{3}{4}. Can you solve it now.
    Spoiler:

    \sin^{-1}x+\sin^{-1}y = \frac{\pi}{3}

    \implies \sin^{-1} x = \frac{\pi}{3} - \sin^{-1} y

    \implies x = sin \left( \frac{\pi}{3} - sin^{-1} y \right)

    \implies x = \frac{\sqrt{3}}{2}\cos \left( sin^{-1} y \right) - \frac{1}{2} y

    \implies x + \frac{1}{2} y = \frac{\sqrt{3}}{2} \sqrt{1- y^2}

    \implies \left( x + \frac{1}{2} y \right)^2 = \frac{3}{4} \left(1- y^2 \right)

    \implies x^2 + y^2 + xy = \frac{3}{4}
    Last edited by kalyanram; September 12th 2012 at 08:28 AM.
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