# Thread: Chain of Sine and Cosine

1. ## Chain of Sine and Cosine

As I was messing around in Maple (yes I'm a nerd), I noticed something.

It appears as if the infinite chain $\sin(\cos(\sin(\cos(\sin(\cos(.....(\sin(\cos(x))) ))))))$ converges for all x.

It appears to be converging to a number very close to $\ln(2)$, but I don't think it actually does, unless it's really, really slowly.

The number (calculated for 7 sine-cosine pairs) is approximately $0.69481969$.

Also, if you let the first term be cosine instead of sine, this chain gets close to $0.768169156736795977462$.

Does anyone know anything about this?

2. Originally Posted by redsoxfan325
As I was messing around in Maple (yes I'm a nerd), I noticed something.

It appears as if the infinite chain $\sin(\cos(\sin(\cos(\sin(\cos(.....(\sin(\cos(x))) ))))))$ converges for all x.

It appears to be converging to a number very close to $\ln(2)$, but I don't think it actually does, unless it's really, really slowly.

The number (calculated for 7 sine-cosine pairs) is approximately $0.69481969$.

Also, if you let the first term be cosine instead of sine, this chain gets close to $0.768169156736795977462$.

Does anyone know anything about this?
This kind of thing is called fixed point iteration. Starting with $x_0$ being whatever, you let $x_1=\sin(\cos x_0)$ and more generally $x_{n+1}=\sin(\cos x_n)$ for every $n\geq 0$.

There are various assumptions that ensure that the sequence converges (for instance, here the function $f(x)=\sin\cos x$ is contracting: $|f(x)-f(y)|\leq k|x-y|$ where $k\in(0,1)$, hence Banach's fixed point theorem applies). Then, if $x_n\to\ell$, necessarily we have $\ell=\sin\cos\ell$ (taking the limit in $x_{n+1}=\sin\cos x_n$).

So that your constants are just the fixed points of $\sin\cos$ and $\cos\sin$: the numbers $c_1,c_2$ such that $\sin\cos c_1=c_1$ and $\cos\sin c_2=c_2$ (they are unique). I don't think there exits a simple explicit formula for them.

You probably can find more about this if you look for "Picard iteration", or "Banach fixed point theorem" with Google.