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Math Help - Complex problem

  1. #1
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    Question Complex problem

    How do I find the answer to e^i(pi/18) *without* using a calculator.

    Thanks
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  2. #2
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    Quote Originally Posted by chancey
    How do I find the answer to e^i(pi/18) *without* using a calculator.

    Thanks
    Use Euler's Formula thus,
    e^{i(\pi/18)}=\cos (\pi/18)+i\sin(\pi/18)
    All you need to do is find what (co)sine of \pi/18 is, you can do this by using the third angle identity,
    <br />
\cos 3\theta=4\cos^3 \theta-3\cos \theta<br />
    Which reduces to solving a cubic equation.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker
    Use Euler's Formula thus,
    e^{i(\pi/18)}=\cos (\pi/18)+i\sin(\pi/18)
    All you need to do is find what (co)sine of \pi/18 is, you can do this by using the third angle identity,
    <br />
\cos 3\theta=4\cos^3 \theta-3\cos \theta<br />
    Which reduces to solving a cubic equation.
    That doesnt help because sine and cosine are made up of the very expression I am trying to solve. This problem has been bugging me for ages
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  4. #4
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    Quote Originally Posted by chancey
    That doesnt help because sine and cosine are made up of the very expression I am trying to solve. This problem has been bugging me for ages
    Since, \cos (\pi/6)=\frac{\sqrt{3}}{2}}
    You have,
    \frac{ \sqrt{3} }{2}=4x^3-3x
    Maybe time later I will solve this cubic, I predict it will look messy.
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  5. #5
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    Ahh, I see. So there is no formula(s) for solving any complex exponential? What if I had something really messy like, e^{0.155i}?

    And yes, when that cubic equation is solved it gives the correct answer
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