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Math Help - Complex Numbers

  1. #1
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    Complex Numbers

    Given that z = cos Ѳ + i sin Ѳ show that
    2/(1 + z) = 1 i tan Ѳ

    How can I solve that? Please help me.
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  2. #2
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    Quote Originally Posted by geton View Post
    Given that z = cos Ѳ + i sin Ѳ show that
    2/(1 + z) = 1 i tan Ѳ

    How can I solve that? Please help me.
    Plug'n'chug:
    \frac{2}{1 + z}

    = \frac{2}{1 + cos(\theta) + i~sin(\theta)}

    = \frac{2}{1 + cos(\theta) + i~sin(\theta)} \cdot \frac{1 + cos(\theta) - i~sin(\theta)}{1 + cos(\theta) - i~sin(\theta)}

    = \frac{2(1 + cos(\theta) - i~sin(\theta))}{(1 + cos(\theta))^2 + sin^2(\theta)}

    = \frac{2(1 + cos(\theta) - i~sin(\theta))}{1 + 2~cos(\theta) + cos^2(\theta) + sin^2(\theta)}

    = \frac{2(1 + cos(\theta) - i~sin(\theta))}{2 + 2~cos(\theta)}

    = \frac{1 + cos(\theta) - i~sin(\theta)}{1 + cos(\theta)}

    = \frac{1 + cos(\theta)}{1 + cos(\theta)} - i~\frac{sin(\theta)}{1 + cos(\theta)}

    = 1 - i~\frac{sin(\theta)}{1 + cos(\theta)}

    = 1 - i~tan \left ( \frac{\theta}{2} \right )

    -Dan
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