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Thread: Complex numbers problem

  1. #1
    Junior Member rednest's Avatar
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    Cool Complex numbers problem

    Using De Moivre's Theorm, express $\displaystyle tan 5\theta$ in terms of $\displaystyle tan \theta$.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by rednest View Post
    Using De Moivre's Theorm, express $\displaystyle tan 5\theta$ in terms of $\displaystyle tan \theta$.
    De Moivre's theorem tells us that:

    $\displaystyle (\cos(\theta)+i \sin(\theta))^5=\cos(5\theta)+i \sin(5\theta)$

    Now I will write $\displaystyle s$ for $\displaystyle \sin(\theta)$ , $\displaystyle c$ for $\displaystyle \cos(\theta)$ and $\displaystyle t$ for $\displaystyle \tan(\theta)$. Then expanding the left hand side and equating real and imaginary parts we get:

    $\displaystyle
    \cos(5\theta)=c^5-10c^3s^2+5c~s^4
    $

    and:

    $\displaystyle
    \sin(5\theta)=5c^4s-10c^2s^3+s^5
    $.

    Therefore:

    $\displaystyle
    \tan(\theta)=\frac{5c^4s-10c^2s^3+s^5}{c^5-10c^3s^2+5c~s^4}
    $.

    Now on the right divide top and bottom through by $\displaystyle c^5$ and you are done

    RonL
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