Lee's Smooth Manifolds Problem 1-8 - Angle function is a smooth coordinate chart. The following is Problem 1-8 in Lee's Introduction to Smooth Manifolds, 2nd Edition:

By identifying $\mathbb R^2$ with $\mathbb C$, we can think of the unit circle $\mathbb S^1$ as a subset of the complex plane. An angle function on a subset $U \subset \mathbb S^1$ is a continuous function $\theta: U \to \mathbb R$ such that $e^{i \theta(z)} = z$. Show that there exists an angle function $\theta$ on an open subset $U$ of $\mathbb S^1$ if and only if $U \neq \mathbb S^1$ For any such angle function, show that $(U, \theta)$ is a smooth coordinate chart for $\mathbb S^1$ with its standard smooth structure.

My question regards the last part: showing that $(U, \theta)$ is a coordinate chart compatible with the standard smooth structure of $\mathbb S^1$.
I have no clue where to start. Any hints will be the most appreciated.
 A: Ok, so Lee defines the coordinate charts as follows (this is in Example 1.2). First, let $U_i^+ = \{(x^1,x^2)\in \mathbb{S}^1\mid x^i > 0\}$, and similarly $U_i^- = \{(x^1,x^2)\in \mathbb{S}^1\mid x^i < 0\}$. Now define $\varphi_i^+:U_i^+\to\mathbb{R}$ as $\varphi_i(x^1,x^2) = x^i$ and $\varphi_i^-:U_i^-\to\mathbb{R}$ as $\varphi_i(x^1,x^2) = x^i$. Our atlas is therefore $\{U_i^\pm, \varphi^\pm\}$. Again, all of this is from Example 1.2.
Now, we are asked to show that for $U$ an open subset of $\mathbb{S}$ and $(U,\theta:U\to \mathbb{R})$ is also a smooth coordinate chart. This is also a chart if it is a homeomorphism, and if it is compatible with the atlas we already have. In otherwords, we want to show that $\varphi_i^\pm\circ\theta^{-1}$ and $\theta\circ (\varphi_i^\pm)^{-1}$ are smooth.
To do this, try writing out what $\theta$ and $\theta^{-1}$ are in terms of coordinates. Try composing them with $\varphi_i^\pm$ or its inverse. Also remember that $\theta\circ (\varphi_i^\pm)^{-1} = (\varphi_i^\pm\circ\theta^{-1})^{-1}$, so it's enough to show that one of them is smooth, and has a smooth inverse. Does this help?
