I have a bit of a basic question as I can't recall exactly what it is. I have the function outlined below and I am unsure what type of equation it is or how to solve it, I just know that it requires a numerical methods to solve. Help with either would be appreciated.
The function is a rearranging of the position function.
$$K = -Acos(\omega t) - Bsin(\omega t_0)t$$
Where K is some known value, A and B are constants, $t_0$ is the initial time which is known and the function variable to solve for is $t$
Thanks.
You are looking for the zero of function $$f(t)=K+A\cos(\omega t) + B\sin(\omega t_0)\,t$$ I would prefer to let $x=\omega t$ and work $$f(x)=K+A\cos(x)+\frac B \omega \sin(\omega t_0)\,x$$ Let $C=\frac B \omega \sin(\omega t_0)$ to make the equation $$f(x)=K+A\cos(x)+Cx$$ If $x$ is supposed to be small, expanding as Taylor series you should get $$f(x)=(A+K)+C x-\frac{A x^2}{2}+O\left(x^3\right)$$ from which an approximation $$x_{est}=\frac{\sqrt{ 2A^2+2 A K+C^2}+C}{A}$$ and, since $C$ is probably very large, approximate again $$x_{est}=\frac{2 C}{A}+O\left(\frac{1}{C^2}\right)$$ Use Newton method to polish the root.