The ring of trigonometric functions over $\mathbb{R}$ is the ring generated by $\sin{x}$ and $\cos{x}$.
What's the reason for why any function $f$ in this ring can be written as $ f(x)=a_0+\sum_{k=1}^n(a_k\cos{kx}+b_k\sin{kx}) $ for $a_0,a_k,b_k\in\mathbb{R}$?