Prove that quotient topology $\mathbb{R}/{\sim}$ is the circle topology.

Question

On $\mathbb{R}$ is given relation:

$ x \sim y \Leftrightarrow (x \bmod \space 1) = (y \bmod \space 1) $

I have to prove that quotient topology $\mathbb{R}/{\sim}$ is the circle topology. I need to build homeomorphism between $R/\sim$ and $S^1$ where $S^1 = \{ (x,y) \in \mathbb{R} , x^2 + y^2 =1 \}$., this map is $q$. And standard quorum map i v. I need to have a function, I thought about $ f \rightarrow S^1, f(x) =(\cos 2\pi x, \sin2\pi x)$. How to show that $f$ is a homeomorphism?