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Thread: Complex Polar Forms, Sin and Cos Angles

  1. #1
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    Complex Polar Forms, Sin and Cos Angles

    Problem: Find all solutions of $\displaystyle z^{3} = -8$.

    I can do the entire problem except for one part. After putting it into the correct form,

    $\displaystyle |z|^{3}(cos3\theta+isin3\theta)$,

    I do not know how to find the values of $\displaystyle cos3\theta$ or $\displaystyle sin3\theta$. I know that $\displaystyle |z|^3$ is 8, but I can't figure out the values of cos and sin.

    Any help is appreciated.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Complex Polar Forms, Sin and Cos Angles

    Quote Originally Posted by tangibleLime View Post
    Problem: Find all solutions of $\displaystyle z^{3} = -8$.

    I can do the entire problem except for one part. After putting it into the correct form,

    $\displaystyle |z|^{3}(cos3\theta+isin3\theta)$,

    I do not know how to find the values of $\displaystyle cos3\theta$ or $\displaystyle sin3\theta$. I know that $\displaystyle |z|^3$ is 8, but I can't figure out the values of cos and sin.

    Any help is appreciated.
    $\displaystyle |z| = 2$.
    So, $\displaystyle cos\ 3\theta+isin\ 3\theta=-1$.
    Equate real and imaginary terms to find $\displaystyle \theta$.
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    Re: Complex Polar Forms, Sin and Cos Angles

    Thanks,

    So since $\displaystyle |z|^3 = 8$, I need $\displaystyle cos3\theta+isin3\theta$ to equal -1 to satisfy the initial equation where $\displaystyle z^3 = -8$. The only way to get -1 from $\displaystyle cos3\theta+isin3\theta$ is to have $\displaystyle cos3\theta = -1$ and $\displaystyle sin3\theta = 0$ to get rid of the imaginary number and return (-1 + i0).

    Correct?
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    MHF Contributor alexmahone's Avatar
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    Re: Complex Polar Forms, Sin and Cos Angles

    Quote Originally Posted by tangibleLime View Post
    Thanks,

    So since $\displaystyle |z|^3 = 8$, I need $\displaystyle cos3\theta+isin3\theta$ to equal -1 to satisfy the initial equation where $\displaystyle z^3 = -8$. The only way to get -1 from $\displaystyle cos3\theta+isin3\theta$ is to have $\displaystyle cos3\theta = -1$ and $\displaystyle sin3\theta = 0$ to get rid of the imaginary number and return (-1 + i0).

    Correct?
    Yes. You should get three values of $\displaystyle \theta$ in the interval $\displaystyle [0,\ 2\pi]$.
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    Re: Complex Polar Forms, Sin and Cos Angles

    Quote Originally Posted by tangibleLime View Post
    Problem: Find all solutions of $\displaystyle z^{3} = -8$.
    Here is some notation: $\displaystyle \exp(i\theta)=\cos{\theta)+i\sin(\theta)$

    So we can write $\displaystyle -8=8\exp(\pi i)$
    The cube roots of that is then $\displaystyle 2\exp \left( {\frac{{i\pi }}{3} + \frac{{2\pi ik}}{3}} \right),~~k=0,1,2$
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