The iteration formula
$$x_{n+1}=x_n-(\cos x_n)(\sin x_n)+R\cos^2x_n$$
where $R$ is a positive constant is obtained by applying Newton's method to some function $f(x)$. What is $f(x)$? What can this formula be used for?
I got the solution for $f(x)$ to be $\dfrac{\sin x}{x\cos x} = \dfrac{1}{x}\tan x$ (I can show the work if needed), but I'm struggling to see what the iteration formula is useful for? I would guess finding roots of $\dfrac{1}{x}\tan x$ at certain locations. Thanks for any help!
Best Answer
If $f(x) = \tan x - R$ then $f'(x) = \sec^2 x$, and plugging these into Newton's method gives $$ x_{n+1} = x_n - \frac{\tan x_n - R}{\sec^2 x_n} = x_n - \sin x_n \cos x_n + R\cos^2 x_n. $$ This would be used to solve $\tan x - R = 0$, or $x = \tan^{-1}(R)$ for a given value of $R$.