-1

$$n * x * cos(\frac{\pi}2 * \frac{x}{x+b}) + c = y$$

How would I get X on one side of the equation instead of y? Normally I work the equation forwards knowing X. The other variables are constants.

for example: $$0.000025 * 1000 * cos(\frac{\pi}2 * \frac{1000}{1000+7000}) + 0.30 = y$$ $$y = 0.32452$$

What id like to know is given y how do i solve for x? $$0.000025 * x * cos(\frac{\pi}2 * \frac{x}{x+7000}) + 0.30 = 0.32452$$ $$x = ?$$ where x > 0

J. Doe
  • 3
  • Are you sure that there is $\cos\left(\frac{\pi}{2}\color{red}{\cdot}\frac{x}{x+b}\right)$ instead of $\cos\left(\frac{\pi}{2}\color{red}{+}\frac{x}{x+b}\right)$? –  Aug 29 '15 at 14:13
  • I am 100% sure. why what does that mean? – J. Doe Aug 29 '15 at 14:26
  • I thought if there was $+$ instead of $\cdot$, you could apply some of the formulas like $\cos(\frac{\pi}{2}+\beta)=\ldots$. But I don't know if it's the proper method to solve that kind of equation. Look here: https://en.wikipedia.org/wiki/List_of_trigonometric_identities –  Aug 29 '15 at 14:28
  • Please consult this tutorial on how to typeset mathematics on this site. As written, your question is difficult to read. – N. F. Taussig Aug 29 '15 at 14:29
  • your third equation has no solutions for $x$ – Dr. Sonnhard Graubner Aug 29 '15 at 15:57
  • is there multiple solutions for x? how is there no solution when I can plug in x = 1000 and get the solution for the 3rd equation? if there were more information could it be solved? – J. Doe Aug 30 '15 at 00:09

1 Answers1

0

When the equation contains at the same time polynomial and trigonometric functions, there is no explicit solution for the root(s). This is already the case for the equation $x=\cos(x)$.

Then, you need to consider numerical methods and probably one of the simplest will be Newton method which, starting from a "reasonable" guess $x_0$, will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Let us play with $$f(x)=0.000025\, x\, \cos(\frac{\pi}2 \, \frac{x}{x+7000}) + 0.30 - 0.34567$$ in order to change from the initial problem (you already know the answer).

If you plot the function over a large range (say from $x=0$ to $x=5000$), you would "see" that there is a root close to $x=2000$. So, let us use $x_0=2000$ and $$f'(x)=0.000025 \cos \left(\frac{\pi x}{2 (x+7000)}\right)-0.000025 x \left(\frac{\pi }{2 (x+7000)}-\frac{\pi x}{2 (x+7000)^2}\right) \sin \left(\frac{\pi x}{2 (x+7000)}\right)$$ and start applying Newton method; this will generate the following iterates $$x_1=1937.90383948004$$ $$x_2=1938.14506264438$$ $$x_3=1938.14506627915$$ which is the solution for fifteen significant figures.

Claude Leibovici
  • 260,315