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Math Help - complex numbers

  1. #1
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    complex numbers

    hey, i have another question, this time to do with complex numbers.

    it asks to find the real part of z=e^(1-i pi/3)

    the way i started doing it was breaking it into e^1 - e^i pi/3
    which gave me e^1 - cos pi/3 + sin pi/3
    which reduces down to (1+sqrt(3))/2 + i/2) + e^1 after using special triangles.

    does that make sense?
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by jackm7 View Post
    hey, i have another question, this time to do with complex numbers.

    it asks to find the real part of z=e^(1-i pi/3)

    the way i started doing it was breaking it into e^1 - e^i pi/3
    which gave me e^1 - cos pi/3 + sin pi/3
    which reduces down to (1+sqrt(3))/2 + i/2) + e^1 after using special triangles.

    does that make sense?
    Your Idea is good, but I think that

    e^{1-\frac{i\pi}{3}}=e^{1}\cdot e^{-\frac{\pi}{3}i}=e(\cos(\pi/3)-i\sin(\pi/3))

    Then just simplify and take the real part
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  3. #3
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    The formulas we can use here are:

    <br />
\begin{aligned}<br />
a^{x+y} &= a^xa^y \\<br />
e^{ix} &= \cos x+i \sin x.\\<br />
\end{aligned}<br />

    Therefore,

    <br />
\begin{aligned}<br />
e^{1-i\frac{\pi}{3}}&=e^1e^{-i\frac{\pi}{3}}\\<br />
&=e\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)\\<br />
&=e\left(\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)\\<br />
&=\frac{1}{2}e-ie\frac{\sqrt{3}}{2}.\\<br />
\end{aligned}<br />

    Edit: Corrected the sign of the second exponent.
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