Function that maps the set of all angles in 360 rotation to the set of all 2D directions?

Question

I am trying to think about a function that maps a 360 rotation to every direction in 2D space.

$0^{\circ} \to (0,1) \\ \vdots \\ 90^{\circ} \to (1,0) \\ \vdots$

Thank you in advance.

Answer 1

$$ \theta \mapsto (\cos{\theta},\sin{\theta}) $$

Answer 2

If you don't mind going via complex space, you can send:

$$\theta\mapsto\left(\mathcal{Re}(e^{i\theta}),\mathcal{Im}(e^{i\theta})\right),$$

although if $\theta$ is given in degrees, you'll want to change it to radians first, i.e., replace $\theta$ with $\frac{\pi\theta}{180}$.