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Math Help - Proof of polar coordinates

  1. #1
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    Proof of polar coordinates

    Let z e C.

    0 is how I write theta below !!!

    If z = r cos(0) + r sin(0) i in polar coordinates, then

    z^-1 = r^-1 cos (-0) + r^-1 sin(-0) i and,

    z^-1 = r^-1 cos (0) - r^-1 sin(0) i


    How do I prove this? Thanks in advance!
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  2. #2
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    Quote Originally Posted by jzellt View Post
    Let z e C.

    0 is how I write theta below !!!

    If z = r cos(0) + r sin(0) i in polar coordinates, then

    z^-1 = r^-1 cos (-0) + r^-1 sin(-0) i and,

    z^-1 = r^-1 cos (0) - r^-1 sin(0) i


    How do I prove this? Thanks in advance!
    This is an application of DeMoivre's Theorem...

    z^n = r^n(\cos{(n\theta)} + i\sin{n\theta}).


    So z^{-1} = r^{-1}[\cos{(-\theta)} + i\sin{(-\theta)}].


    These are some well-known identities:

    \cos{(-\theta)} = \cos{\theta} and \sin{(-\theta)} = -\sin{\theta}

    (Check their graphs if you have to)...


    So z^{-1} = r^{-1}(\cos{(-\theta)} + i\sin{-\theta}) = r^{-1}(\cos{\theta} - i\sin{\theta}).
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