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Math Help - Expressing a complex number in cartesian form

  1. #1
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    Expressing a complex number in cartesian form

    Im stuck on a textbook question that has no previous examples. Express  \left(\frac{\sqrt{3}-i}{2}\right)^{101} in cartesian form.

    OK. So i know De moivre's Theorem but not sure if i can use it here (and thats going away from Cartesian form too)

    But could I say that here the form r(cos x + isin x) can be deduced because \frac{\sqrt{3}}{2} is cos\frac{\pi}{6} and then deduce sin.....

    But not sure where to go from here!...
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  2. #2
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    Re: Expressing a complex number in cartesian form

    Quote Originally Posted by FelixHelix View Post
    Im stuck on a textbook question that has no previous examples. Express  \left(\frac{\sqrt{3}-i}{2}\right)^{101} in cartesian form.

    OK. So i know De moivre's Theorem but not sure if i can use it here (and thats going away from Cartesian form too)

    But could I say that here the form r(cos x + isin x) can be deduced because \frac{\sqrt{3}}{2} is cos\frac{\pi}{6} and then deduce sin.....

    But not sure where to go from here!...
     z = \left(\frac{\sqrt 3}{2}-\frac{1}{2}i\right)^{101}=\left(\cos \frac{11 \pi}{6}+i\sin\frac{11 \pi}{6}\right)^{101}

     z= \cos \frac{1111 \pi}{6}+i\sin\frac{1111 \pi}{6} = \cos \frac{7 \pi}{6}+i\sin\frac{7 \pi}{6}

     z = -\frac{\sqrt 3}{2}-\frac{1}{2}i
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  3. #3
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    Re: Expressing a complex number in cartesian form

    Quote Originally Posted by FelixHelix View Post
    Im stuck on a textbook question that has no previous examples. Express  \left(\frac{\sqrt{3}-i}{2}\right)^{101} in cartesian form....
    \frac{\sqrt{3}-i}{2}=2\exp\left(\frac{-\pi}{6}\right)
    Now apply De moivre's Theorem.
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  4. #4
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    Re: Expressing a complex number in cartesian form

    I'm confused by the workings here. Could you explain how you get the original argument here?
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  5. #5
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    Re: Expressing a complex number in cartesian form

    HI Plato, Im confused, how come your workings are different to Princeps? and how do you get r = 2 in your polar form here?
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  6. #6
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    Re: Expressing a complex number in cartesian form

    Quote Originally Posted by FelixHelix View Post
    HI Plato, Im confused, how come your workings are different to Princeps? and how do you get r = 2 in your polar form here?
    They are not different; they are equivalent. You should know that.
    I just prefer the principal argument, -\pi<\theta\le\pi.
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