One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does
$$ x_0 = \sin 1, \space x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ and after a while, $$ x = \pi $$
I know this to be true because I have evaluated this on my TI-84 and more deeply evaluated it with this program I made: https://repl.it/@RobertoBean/Pi-Evaluator To $100000$ Iterations (which I believe is enough)
So what's the math behind it? Why does adding $\sin 1$ in this manner produce $\pi$? Why doesn't doing the same thing using $100$ produce $\pi$? For example, $$\sin(1) = 0.841470...,\space \sin(1) + \sin(0.8414...), \sin(1) + \sin(0.841470...) + \sin(1.587095126...),\space ... $$ $$ = \pi$$
My question is different from the mentioned because my function is not taking the sin of a sin consecutively, but is this following function: $f(x) = x + \sin x$ and is not $f(x) = \sin x$ and therefore can exhibit different properties I need explained.