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.

 

[CellCoree]

I would approach this problem by first understanding the limitations of non-closed form equations. These types of equations do not have a specific solution and therefore require approximation methods to find an approximate solution. In this case, the equation contains trigonometric functions which can be challenging to solve for a specific value.

One approach to approximate the value of b for a given value of c would be to use numerical methods such as iteration or interpolation. This involves plugging in different values of b into the equation and checking for convergence to a certain value. This method may require multiple iterations to achieve the desired level of accuracy.

Another approach would be to use a graphing calculator or software to plot the equation and visually estimate the value of b. This method may not be as accurate but can provide a rough estimate.

Additionally, it may be helpful to consider the range of values for b and c and how they affect the overall equation. This can provide insights into which values of b may be more likely to provide a close approximation to the equation.

In conclusion, approximation of values from non-closed form equations requires a combination of numerical methods, visualization, and understanding of the underlying mathematical principles. With careful consideration and experimentation, an accurate approximation of b for a given c value can be obtained.