Exercise: Show that spherical coordinates form a slice chart for $\mathbb{S}^2$ in $\mathbb{R}^3$ on any open subset where they are defined.
Solution: For this post I will just show this for one portion of the sphere, for brevity. $(U, \psi)$ is a spherical coordinate chart on $\mathbb{R}^3$ where $U = (0, \infty) \times (0, \pi) \times (-\pi, \pi)$ and $\psi(\rho, \varphi, \theta) = (\rho\cos\theta \sin\psi, \rho\sin\theta\sin\varphi, \rho\cos\varphi)$.
We know that $\mathbb{S}^2 \cap U$ is just the portion of the sphere where $y > 0$ (in Cartesian coordinates). So, $\psi(\mathbb{S}^2 \cap U) = \{(\cos\theta\sin\varphi, \sin\theta\sin\varphi, \cos\varphi): -\pi < \theta < \pi, 0 < \varphi < \pi\}$. This isn't a $2$-slice of $\psi(U)$ because none of the coordinates are constant. Where am I going wrong?
What you wrote down is $\psi(U) \cap \mathbb S^2$, not $\psi(U\cap \mathbb S^2)$. The spherical coordinate map is $\psi^{-1}$, not $\psi$.