I'm struggling to find the cosets of $\langle\mathbb{R}, +\rangle / \langle 360\mathbb{Z}, +\rangle$.
Usually I only find cosets of finite groups, in which case I know the cardinality of the subset and then divide the groups cardinality by that to get the number of cosets. Then it is just a matter of finding what elements are in the other cosets.
But I can't figure out what it is for the infinite sets above. How many cosets are there and how do you know? At the very least, I can imagine the subgroup would be $\{0, 360, 720, ...\}$ (because it has to be a subgroup and have the identity).
The cosets are of the form $r+H$ for $r$ in the interval $[0, 360)$ and $H:=360\Bbb Z$. Here's an example for illustration:
$$\pi+H=\{\pi+h\mid h\in 360\Bbb Z\}.$$
That's about as concise as I can get it, prima facie.
Further: $$\pi+H=\{\dots , \pi-720, \pi-360, \pi, \pi+360, \pi+720, \dots\}.$$
It might help to note that two cosets $a+H$ and $b+H$ are the same if and only if $a$ and $b$ differ by an element of $H$; that is, if and only if $a-b\in 360\Bbb Z$.