I am new to root-finding and using numerical methods.
Using Newton's method below: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_0)}$$ using this chord formula where the chord length $c$ is 1 cm: $$c=2r\sin\frac{\theta}{2}$$ supposing the radius is 1.1 cm and the angle θ is unknown, show the iterative Newton's Method equation you would use to find an approximate value for $\theta$ in the context of this problem (using the appropriate function and derivative).
I have tried rearranging it to $\theta$ but Symbolab gives me two solutions which add an $n$ to the equation. I found this equation but not sure if it right:$$\theta = 2\arcsin\frac{c}{2r}$$
But when I derive this, it becomes zero so I'm unsure if my equation is right either. I would just like help deriving this equation and then applying it to the problem.
You are expected to work with $$ f(x)=c-2r\sin\frac x2\implies f'(x)=-r\cos\frac x2 $$ and indeed this equation has multiple solutions. You need to pick the one in the correct interval, what that is comes from additional knowledge about the geometrical situation. One could for instance ask for the smallest positive solution, which will be inside the interval $[0,\pi]$ if $0<c<2r$.