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Complex Numbers - Complex Roots of Unity

  1. Jan 3, 2012 #1
    Need help with this please:

    1. The problem statement, all variables and given/known data
    (1 + cosθ + isinθ) / (1 - cosθ - isinθ) = icotθ/2

    The first step in the solutions shows:

    (2cos^2θ/2 + i2sinθ/2cosθ/2) / (2sin^2θ/2 - i2sinθ/2cosθ/2)

    2. Relevant equations

    I can't get there.

    3. The attempt at a solution

    I tried multiplying by: (1 - cosθ + isinθ) / (1 - cosθ + isinθ), with no luck.

    the top line of my attempt shows i2sinθ + i2sinθcosθ
    because 1-cos^2θ=sin^2θ


    my signs could be wrong but still. how does the θ become θ/2. this is doing my head in. maybe just sleep on it.
     
    Last edited: Jan 3, 2012
  2. jcsd
  3. Jan 3, 2012 #2

    Curious3141

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    Homework Helper

    OK, you know from the standard double-angle formulae that [itex]\sin{2\alpha} = 2\sin\alpha\cos\alpha[/itex] and that [itex]\cos{2\alpha} = 2\cos^2{\alpha} - 1 = 1 - 2\sin^2{\alpha}[/itex].

    Now put [itex]\alpha = \frac{\theta}{2}[/itex].

    The results you get are often called the half-angle formulae, but it's not worth remembering them specifically because they're so easily derived from the double-angle formulae.
     
  4. Jan 3, 2012 #3
    Thanks for the prompt reply and pointing that out!

    So simple now i can see that.
     
  5. Jan 3, 2012 #4

    Curious3141

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    Homework Helper

    You're welcome.
     
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