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Thread: Ugh, blanking out --- complex numbers, exponents...

  1. #1
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    Ugh, blanking out --- complex numbers, exponents...

    solve $\displaystyle (1-i)^{2009}$

    Attempt:

    r = $\displaystyle \sqrt{2}$

    tan$\displaystyle \Theta$ = -1 $\displaystyle \rightarrow$ tan$\displaystyle \Theta$ = $\displaystyle \frac{7\pi}{4}$

    so...

    $\displaystyle (1-i)^{2009}$ = $\displaystyle \sqrt{2}^{2009}(cis \frac{2009 \cdot 7\pi}{4})$

    $\displaystyle \sqrt{2}^{2009} = \sqrt{2} \cdot 2^{1004} $

    so... $\displaystyle (1-i)^{2009}$ = $\displaystyle \sqrt{2} \cdot 2^{1004}(cis \frac{2009 \cdot 7\pi}{4}) $

    can someone please tell me what I do with $\displaystyle (cis \frac{2009 \cdot 7\pi}{4}) $

    Thanks. I'm blanking out...
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Try to compute first:

    (1-i)^2
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  3. #3
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    Try to compute first:

    (1-i)^2
    I know the question is much easier and quicker that way, but I want to try and solve the question with De Mouvre's (or wtvr his name is) equation.
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  4. #4
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    Quote Originally Posted by jayshizwiz View Post
    solve $\displaystyle (1-i)^{2009}$

    Attempt:

    r = $\displaystyle \sqrt{2}$

    tan$\displaystyle \Theta$ = -1 $\displaystyle \rightarrow$ tan$\displaystyle \Theta$ = $\displaystyle \frac{7\pi}{4}$

    so...

    $\displaystyle (1-i)^{2009}$ = $\displaystyle \sqrt{2}^{2009}(cis \frac{2009 \cdot 7\pi}{4})$

    $\displaystyle \sqrt{2}^{2009} = \sqrt{2} \cdot 2^{1004} $

    so... $\displaystyle (1-i)^{2009}$ = $\displaystyle \sqrt{2} \cdot 2^{1004}(cis \frac{2009 \cdot 7\pi}{4}) $

    can someone please tell me what I do with $\displaystyle (cis \frac{2009 \cdot 7\pi}{4}) $

    Thanks. I'm blanking out...
    Note that $\displaystyle \frac{2009 \cdot 7\pi}{4} = 3514 \pi + \frac{7 \pi}{4} $ ....
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  5. #5
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    Write the given problem as

    $\displaystyle (1-i)^{2009} = (\sqrt{2})^{2009}[cos(-\frac{\pi}{4}) + sin(-\frac{\pi}{4})]^{2009}$

    = $\displaystyle (\sqrt{2})^{2009}[cos(-\frac{2009\pi}{4}) + sin(-\frac{2009\pi}{4})]$

    =$\displaystyle (\sqrt{2})^{2009}[cos(-\frac{(2008 + 1)\pi}{4}) + sin(-\frac{(2008 + 1)\pi}{4})]$

    = $\displaystyle (\sqrt{2})^{2009}[cos(-\frac{\pi}{4}) + sin(-\frac{\pi}{4})]$

    Now proceed.
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