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Thread: Solve the equation of Cos3(theta) = 1/2

  1. #1
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    Solve the equation of Cos3(theta) = 1/2

    Solve the equation of Cos3(theta) = 1/2 , for theta is an element of real numbers, where (theta) is in radius.

    I feel like it's simple but I don't know how :-/
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    Re: Solve the equation of Cos3(theta) = 1/2

    $\displaystyle cos(3\theta) = \frac{1}{2}$

    $\displaystyle 3\theta = cos^{-1}(\frac{1}{2})$

    $\displaystyle 3\theta =60 $ degrees

    $\displaystyle \theta = 20$ degrees $\displaystyle + 360k $ where k is any integer.
    Last edited by Goku; Jun 10th 2012 at 10:58 AM.
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    Re: Solve the equation of Cos3(theta) = 1/2

    Quote Originally Posted by DeeRose View Post
    Solve the equation of Cos3(theta) = 1/2 , for theta is an element of real numbers, where (theta) is in radius.
    Is this supposed to be $\displaystyle \cos3\theta=\frac12$ or $\displaystyle \cos^3\theta = \frac12\mathrm?$ If the latter, are you looking for an exact solution?
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    Re: Solve the equation of Cos3(theta) = 1/2

    Quote Originally Posted by Goku View Post
    $\displaystyle \theta = 20$ degrees
    Your procedure is good, but this is only one of infinitely many solutions.
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    Re: Solve the equation of Cos3(theta) = 1/2

    It's Cos3(theta) ..Not Cos^3(theta) Is Goku's response the right one then?
    Last edited by DeeRose; Jun 10th 2012 at 11:29 AM.
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    Re: Solve the equation of Cos3(theta) = 1/2

    Quote Originally Posted by DeeRose View Post
    It's Cos3(theta) .. Is Goku's response the right one then?
    Note that cosine is equal to $\displaystyle \frac12$ for any angle that is coterminal with $\displaystyle \frac\pi3$ or $\displaystyle -\frac\pi3$. Thus

    $\displaystyle \cos3\theta=\frac12$

    $\displaystyle \Rightarrow3\theta = 2k\pi\pm\frac\pi3$

    $\displaystyle = \frac{(6k\pm1)\pi}3$

    $\displaystyle \Rightarrow\theta = \frac{(6k\pm1)\pi}9$

    where $\displaystyle k\in\mathbb{Z}$.
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  7. #7
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    Re: Solve the equation of Cos3(theta) = 1/2

    Thanks alot! :-)
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