Approximation of values from non-closed form equation.

  1. Hello everyone, I'm working on a problem and it turns out that this equation crops up:

    [tex]1 = cos^{2}(b)[1-(c-b)^{2}][/tex]

    where

    [tex]c > \pi[/tex]

    Now I'm pretty sure you can't solve for b in closed form (at least I can't), so what I need to do is for some value of c, approximate the value of b to about 5-6 digits of accuracy. I just need tips to head in the right direction. Anything will be useful. Thank you!
     
  2. jcsd
  3. http://en.wikipedia.org/wiki/Newton's_method

    Put simply, ##x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}##. Just iterate the formula a few times to get an approximate answer.
     
  4. [tex]1 = cos^{2}(b)[1-(c-b)^{2}][/tex]
    [tex]1-cos^{2}(b) = cos^{2}(b)[-(c-b)^{2}][/tex]
    [tex]sin^{2}(b) = cos^{2}(b)[-(c-b)^{2}][/tex]
    [tex]tan^{2}(b) = -(c-b)^{2}[/tex]
    For real solution, positive term = negative term is only possible if they are =0.
    Hence the solution is : [tex]c=b=n\pi[/tex]
     
    1 person likes this.
  5. Oh okay, thanks JJacquelin, I didn't even think to do this.
     
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