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:
Are there any other well-known periodic functions?
"Are there any other well-known periodic functions?"
In one sense, the answer is "no". Every reasonable periodic complex-valued function $f$ of a real variable can be represented as an infinite linear combination of sines and cosines with periods equal the period $\tau$ of $f$, or equal to $\tau/2$ or to $\tau/3$, etc. See Fourier series.
There are also doubly periodic functions of a complex variable, called elliptic functions. If one restricts one of these to the real axis, one can find a Fourier series, but one doesn't do such restrictions, as far as I know, in studying these functions. See Weierstrass's elliptic functions and Jacobi elliptic functions.
There are some unreasonable ones that are periodic.
$$f(x) =\sum_{n=-\infty}^\infty (-1)^{n}e^{-(x-n)^2}$$
is periodic with period 2. Looks like cosine but is not.
This is an extension of David Mitra answer.
Pick any functions which is smooth on an open interval containing $[0,1]$.
Define $g: [0,1] \rightarrow R$ by $g(x)=x^2(1-x)^2f(x)$. Then $g(0)=g'(0)=g(1)=g'(1)=0$.
You can now prove that $h: R \rightarrow R$ defined by
$$h(x) = g( \{ x \}) \,,$$
where $\{ x \}$ represents the fractional part of $x$ is smooth periodic with period 1...
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 $2\pi$ $\forall n\in\mathbb{R}^+$ .
