# Approximation of values from non-closed form equation.

• Legaldose
In summary, the conversation discusses an equation that involves the variables b and c, where c is greater than pi. It is mentioned that the equation cannot be solved for b in closed form and the desired solution is an approximation to 5-6 digits of accuracy. The method of approximation using Newton's method is suggested and further explained. A simpler method is also suggested by setting the positive and negative terms equal to 0, leading to the solution c=b=nπ.
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!

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

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

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.

## 1. What is the definition of approximation?

Approximation is the act of finding a value that is close enough to the exact value of a quantity, especially when the exact value cannot be determined or is too complicated to obtain.

## 2. How is approximation used in scientific research?

Approximation is often used to simplify complex equations or models in order to make them more manageable and easier to understand. It is also used to estimate values that are difficult or impossible to measure directly.

## 3. What is a non-closed form equation?

A non-closed form equation is an equation that does not have a finite solution and cannot be written in a simple, mathematical form. These equations often involve complex functions or unknown variables that cannot be solved for directly.

## 4. Why is it important to approximate values from non-closed form equations?

Approximating values from non-closed form equations allows scientists to still make meaningful predictions and analyze data even when an exact solution is not possible. It also allows for simplification and understanding of complex systems or phenomena.

## 5. What are some methods for approximating values from non-closed form equations?

Some common methods for approximation include using numerical methods, such as iteration or interpolation, or using simplifying assumptions to reduce the complexity of the equation. Another approach is to use statistical techniques, such as regression analysis, to estimate values based on a set of data points.

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