Axiomatic definition of sin and cos?
To quote a previous answer of mine:
Robison, "A new approach to circular functions, π, and lim sin(x)/x", Math. Mag. 41.2 (March 1968), 66–70 [jstor].
In this paper it is shown that the addition law for cosine (and a couple other simple assumptions) uniquely determines cosine and sine.
(This is the paper I cite most often on StackExchange.)
If you don't have access to jstor (and don't want to sign up for their free 3-papers-at-a-time deal), you could try this other answer of mine to a closely related question about exponentiation, in which I adapted Robison's proof to give the following functional characterization of sine and cosine:
Proposition 1. Suppose $C,S\colon\mathbb R\to\mathbb R$ satisfy these conditions:
- $C$ and $S$ are continuous;
- $C(u-v) = C(u)C(v)+S(u)S(v)$ for all $u,v\in\mathbb R$;
- $S(u-v) = S(u)C(v)-C(u)S(v)$ for all $u,v\in\mathbb R$;
- $C$ and $S$ are not both identically zero.
Then there exists $\lambda\in\mathbb R$ such that
$$ C(u) = \cos(\lambda u) \quad\text{and}\quad S(u) = \sin(\lambda u) \text{ .} $$
You'll note that I just took continuity as an axiom, which was convenient and natural in the context of that other question; Robison proves continuity from weaker but less intuitive axioms — if memory serves, the key one is that cosine has a smallest positive root, i.e., there exists $p>0$ such that $C(p)=0$ and $C(x)\ne0$ for $0 < x < p$ — which might serve your purposes better.
(Oh, and since there's some question about degrees and radians and all that: Robison takes the normalization $p=1$, proves that $\lim_{x\to 0}\frac{S(x)}{x}$ exists, then defines $\pi = 2\lim_{x\to 0} \frac{S(x)}{x}$, and then defines $\sin(x) = S(\frac{2x}{\pi})$, and similarly for $\cos$. This manoeuvre amounts to defining $\pi$ by the condition $\lim_{x\to 0}\frac{\sin x}{x} = 1$, very much like we sometimes define $e$ by the condition $\lim_{x\to 0}\frac{e^x-1}{x} = 1$.)
From what you do not want that pretty much leaves functional equations.
There seems to be a system of two functional equations for sine and cosine: (link)
$$ \Theta(x+y)=\Theta(x)\Theta(y)-\Omega(x)\Omega(y) \\ \Omega(x+y)=\Theta(x)\Omega(y)+\Omega(x)\Theta(y) $$