Suppose I have two points on a unit sphere whose spherical coordinates are $A(\theta_1,\phi_1)$ and $B(\theta_2,\phi_2)$, what is the angle between $xy$-plane and arc $AB$?
Maybe I can draw a triangle with the north pole and use cosine rule, but I couldn't figure out how.
Edit: For my application, the angle I need is the minmum angle between the tangent of any point on the arc and $xy$-plane.
Let $\mathbf{a}= (\sin \theta_{1} \cos \phi_{1},\sin \theta_{1} \sin \phi_{1},\cos \phi_{1})$ and $\mathbf{b}=(\sin \theta_{2} \cos \phi_{2},\sin \theta_{2} \sin \phi_{2},\cos \phi_{2})$
The required angle is $$\cos^{-1} \left( \frac{\mathbf{a} \times \mathbf{b} \cdot \mathbf{k}} {|\mathbf{a} \times \mathbf{b}|} \right)$$
The computation of $\mathbf{a} \times \mathbf{b} \cdot \mathbf{k}$ is easy but $|\mathbf{a} \times \mathbf{b}|$ is awful unless $\theta_{1}=\theta_{2}$ or $\phi_{1}=\phi_{2}$.