Are sin and cos the only continuous and infinitely differentiable periodic functions we have? Sin and cos are everywhere continuous and infinitely differentiable.  Those are nice properties to have.  They come from the unit circle.
It seems there's no other periodic function that is also smooth and continuous.  The only other even periodic functions (not smooth or continuous) I have seen are:


*

*Square wave

*Triangle wave

*Sawtooth wave


Are there any other well-known periodic functions?

Of course not.
For example, $\sin_{[n]}(x)$ as shown in http://en.wikipedia.org/wiki/Functional_square_root is in fact a smooth periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+$ .
