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Math Help - Finding Cartesian form from argument and modulus

  1. #1
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    Finding Cartesian form from argument and modulus

    The question:
    Find the "a + ib" form of the complex numbers whose modulus and principle argument are:

    |z| = 3, arg(z) = Pi/8

    My attempt:

    3(cos(\frac{\pi}{8}) + isin(\frac{\pi}{8}))

    I'm certain they want the solution as an exact value, so Pi/8 is going to be a pain to find. I'm thinking I have to use the double-angle formulas, but I'm doing something wrong...

    cos(2x) = cos^2x - sin^2x
    cos (\frac{\pi}{8}) = cos^2(\frac{\pi}{16}) - sin^2(\frac{\pi}{16})

    Clearly that isn't any easier. Have any suggestions? Thanks.
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  2. #2
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    You'll actually need the half-angle formulas

    \sin{\frac{\theta}{2}} = \pm \sqrt{\frac{1 - \cos{\theta}}{2}}

    and

    \cos{\frac{\theta}{2}} = \pm \sqrt{\frac{1 + \cos{\theta}}{2}}.


    In this case, \theta = \frac{\pi}{4}.
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  3. #3
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    Heh, I didn't realise half-angle formulas existed! Thanks!
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