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Analytic Solutions to a Few Trig Equations

  1. Sep 20, 2008 #1
    Is there an analytic solution to an equation of the following form?

    A*cos(w*t) + B*t = C

    where A, B, C, and w are constants

    Maybe it can be solved by expanding the cos() to a series?

    I am also wondering the same question about the following, though I believe that I've read/been told that there is no known analytic solution.

    A*cos([tex]\Theta[/tex]) + B*sin([tex]\Theta[/tex]) = C
  2. jcsd
  3. Sep 20, 2008 #2
    If what you mean by an analytical solution is a finite expression using only "elementary functions" then I don't believe the first has an analytical solution. You could of course always define a solution set:
    [tex]S = \{t | A\cdot \cos(wt) + Bt = C\}[/tex]
    which I would consider a solution, though it doesn't tell us how to solve it.

    For the second it's pretty easy. Rearrange:
    [tex]A \cdot \cos(\Theta) = C - B\cdot \sin (\Theta)[/tex]
    [tex]A^2 (1-\sin^2(\Theta)) = C^2 + B^2 \sin^2(\Theta) - 2BC\cdot \sin(\Theta)[/tex]
    Then it's a simple quadratic equation in [tex]\sin(\Theta)[/tex].
  4. Sep 20, 2008 #3

    The solution to the second is so simple, I almost can't believe I didn't come up with it. I guess that shows what happens when you haven't had a math class in a few years.

    By analytic solution, I mean an equation solved for t, instead of a numerical method.
  5. Sep 20, 2008 #4


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    Don't think there is a closed form solution. You'll have to approximate it numerically.
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