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Math Help - Equations in Polar Coordinates

  1. #1
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    Equations in Polar Coordinates

    Write the following equation in polar coordinates (using t for ):

    (x2 + y2)2 = x2 - y2 becomes



    ____?____ - cos(2t) = 0


    I know that r2=x2+y2 but i'm not sure where everything else leads me.
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  2. #2
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    Quote Originally Posted by mickles View Post
    Write the following equation in polar coordinates (using t for ):

    (x2 + y2)2 = x2 - y2 becomes



    ____?____ - cos(2t) = 0


    I know that r2=x2+y2 but i'm not sure where everything else leads me.
    x^2+y^2=r^2

    x=r\cos(t)

    y=r\sin(t)

    Try this subs and see what happens
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    Quote Originally Posted by TheEmptySet View Post
    x^2+y^2=r^2

    x=r\cos(t)

    y=r\sin(t)

    Try this subs and see what happens
    Okay i did this:

    r^2 = r^2cos^2(x)+r^2sin^2(x)
    1 = cos^2(x) +sin^2(x)
    So 1 = cosx + sinx
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    I also inserted x and y into the equation in my first post and i got this:

    cos^2(x)-sin^2(x) = (r^2cos^4(x)+2r^2cos^2(x)sin^2(x)+r^2sin^4(x))
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  5. #5
    Behold, the power of SARDINES!
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    Quote Originally Posted by mickles View Post
    Write the following equation in polar coordinates (using t for ):

    (x2 + y2)2 = x2 - y2 becomes



    ____?____ - cos(2t) = 0


    I know that r2=x2+y2 but i'm not sure where everything else leads me.
    Remember that \cos^2(t)-\sin^2(t)=\cos(2t)

    so we get

    (r^2)^2=r^2\cos^2(t)-r^2\sin(t) \iff r^4=r^2(\cos^2(t)-\sin^2(t))

    r^4=r^2(\cos(2t)) \iff r^2=\cos(2t) \iff r^2-\cos(2t)=0
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  6. #6
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    Quote Originally Posted by TheEmptySet View Post
    Remember that \cos^2(t)-\sin^2(t)=\cos(2t)

    so we get

    (r^2)^2=r^2\cos^2(t)-r^2\sin(t) \iff r^4=r^2(\cos^2(t)-\sin^2(t))

    r^4=r^2(\cos(2t)) \iff r^2=\cos(2t) \iff r^2-\cos(2t)=0
    Oh okay i understand now. I forgot the identity where cos^2(2t)-sin^2(2t) = cos(2t)

    Thank you very much i understand now!
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