Showing that the points of the unit circle have the same infinity of elements as $\mathbb{R}$ I know that using Bernstein's theorem, I simply need to find two one-to-one functions that map from the unit circle to $\mathbb{R}$ and vice versa. 
I thought something like $f(x, y)=\arctan\big(\frac{x}{y}\big)$ could work as a mapping from the unit circle to $\mathbb{R}$, but it seems like isn't one-to-one since and there seems to be problems when $y=0$. I also really can't think of a one-to-one function that maps from the reals to the unit circle.
Is there a better way to do this problem?

Here’s one that I ought to have known when I first started teaching Calculus, but didn’t till several years later:
$$
t\mapsto\left(\frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2}\right)\,.
$$
For $t\in\Bbb R$, it hits all points on the unit circle save $(0,-1)$. I’m sure you know that it’s not possible to make a bicontinuous map between the real line and the circle. But you can take the displayed formula above and hoke it up with the right discontinuous function to make it a fully one-to-one correspondence.
