Wikipedia states that : In rough terms, a Lie group is a continuous group: it is a group whose elements are described by several real parameters. As such, Lie groups provide a natural model for the concept of continuous symmetry, such as rotational symmetry in three dimensions.
Given the above definition, is it possible to express a helix as a lie group? I am interested in the cosine wave but want to be able to distinquish algebraically between $0^\circ...360^\circ$ and $360^\circ...720^\circ$ and onwards....
Topologically a helix is just a line, so it would necessarily be isomorphic to the reals under addition as a Lie group (since this is the only connected noncompact Lie group up to isomorphism).
Explicitly, consider the helix $H=\{(e^{i\omega t},t)\mid t\in\mathbb{R}\}\subset\mathbb{C}\times\mathbb{R}$ (where $\omega$ is the angular velocity). We can use the multiplication from the cylinder $S^1\times\mathbb{R}$ (where the operation in $\mathbb{R}$ is actually addition, but in $S^`$ is multiplication of complex numbers) and that makes the helix a Lie subgroupof the cylinder $H<S^1\times\mathbb{R}$.
The isomorphism $(\mathbb{R},+)\to H$ is given by $t\mapsto (e^{i\omega t},t)$ of course.
There is also a universal covering map $H\to S^1$ given by $(w,z)\mapsto w$, so for every rotation (interpreted as a phasor, or unit complex number) there are infinitely many elements of the helix which represent it. In particular, for any angle $\theta$ and integer $n$, the elements $\theta+(360^{\circ})n$ are all distinct, interpreted as elements of the helix anyway.
A similar thing can be done in any number of dimensions. The rotation group $\mathrm{SO}(n)$ has a double cover (there isn't anything bigger than a double cover after $n>2$) called the spin group $\mathrm{Spin}(n)$. I call its elements "spins," which can be interpreted as rotations but with a "memory" of how they are performed (up to homotopy), so a $0^{\circ}$ spin is inequivalent to a $360^{\circ}$ spin, but (oddly enough) turns out to be the same as a $720^{\circ}$ spin! This can be illustrated by the so-called "belt-trick" or "dirac string trick" etc. which you can try out with household items or even your own hand!