Convergence of cos, sin, tan functions

Solution 1:

This is explained by stability theory.

The fixed point of $cos$ is the so called Dottie number. The derivative of $cos$ at this number is strictly less than $1$, so the Dottie number is a stable point. With $sin$ and $tan$ the derivative at $0$ equals $1$, so you need to do a more in-depth analysis. The fixed point of $sin$ turns out to be an attractor as well, whereas the dynamics can get quite wild with $tan$. To see what is happening here, it is best you plot a graph of these functions, together with the line $y = x$.

Solution 2:

I think what you do is to find the invariant point function of $\sin x$, $\cos x$ and $\tan x$. In other words, $x=0$ is the root for $\sin x=x$ and $x=0.739085$ is the root for $\cos x=x$. And that is why you can do it iteratively.