Iterated trigonometric functions are functions built by using trigonometric functions recursively. The first useful one I stumbled upon was the iterated sine function, which I defined as follows:

${\displaystyle \sin _{0}x=x}$

${\displaystyle \sin _{1}x=\sin(x)}$

${\displaystyle \sin _{i}x=\sin(\sin _{i-1}(x))}$

This function is interesting in that it arbitrarily approaches a continuous and smooth square wave, without any ringing artifacts, if you append to it a normalization factor as i tends to infinity (this is important otherwise it'll converge to zero).

I was wondering if I could approach other primitive waveforms (such as triangle and sawtooth waves) with a similar method. I found that ${\displaystyle \tan(\sin(x))}$ is close to a smooth triangle wave, but I couldn't manage to make it arbitrarily close. ${\displaystyle \sin(\tan(x))}$ looks like a sawtooth wave, but it gets nasty as the tangent goes to infinity.

Interestingly, iterating ${\displaystyle \tan(\sin(x))}$ 7 times gets pretty close of the square sine function, but it still doesn't get arbitrarily close to it.

On the other hand, iterating cosine and tangent alone gave uninteresting results.