Approximation of values from non-closed form equation.

[Legaldose]

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

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

where

$$c > \pi$$

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!

 

[scurty]

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.

 

[JJacquelin]

$$1 = cos^{2}(b)[1-(c-b)^{2}]$$
$$1-cos^{2}(b) = cos^{2}(b)[-(c-b)^{2}]$$
$$sin^{2}(b) = cos^{2}(b)[-(c-b)^{2}]$$
$$tan^{2}(b) = -(c-b)^{2}$$
For real solution, positive term = negative term is only possible if they are =0.
Hence the solution is : $$c=b=n\pi$$

 

[Legaldose]

Oh okay, thanks JJacquelin, I didn't even think to do this.