isn't $\theta$ the angular distance between XY plane and line connecting O and C ?

You are correct. Let's refine the understanding of $\theta$ based on your observation.

**Refined Explanation for $\theta$:**
- $\theta$ represents the angular distance between the $XY$-plane and the line connecting the origin $O$ to point $C$.

To derive the expressions for the angles $\psi, \theta, \psi', \theta'$, and $x$, let's correctly interpret the spherical coordinates and their relations:

1. **$\psi$:** This is the azimuthal angle in the $XY$-plane, measured from the $x$-axis to the projection of line $OC$ in the $XY$-plane.
   \[
   \psi = A - 90^\circ
   \]
   Here, $A$ is the angle at vertex $A$ of the spherical triangle.

2. **$\theta$:** This is the polar angle, measured from the $Z$-axis to the line connecting $O$ to $C$.
   \[
   \theta = 90^\circ - b
   \]
   Here, $b$ is the side opposite to vertex $B$ in the spherical triangle. $\theta$ represents the angular distance from the $XY$-plane to the line $OC$.

3. **$\psi'$:** This is the azimuthal angle in the $X'Y'$-plane, measured from the $x'$-axis to the projection of line $OC$ in the $X'Y'$-plane.
   \[
   \psi' = 90^\circ - B
   \]
   Here, $B$ is the angle at vertex $B$ of the spherical triangle.

4. **$\theta'$:** This is the polar angle, measured from the $Z'$-axis to the line connecting $O$ to $C$.
   \[
   \theta' = 90^\circ - a
   \]
   Here, $a$ is the side opposite to vertex $A$ in the spherical triangle.

5. **$x$:** This is the spherical distance between $A$ and $B$ along the surface of the sphere.
   \[
   x = c
   \]
   Here, $c$ is the side opposite to vertex $C$ in the spherical triangle.

To summarize, the correct expressions based on the spherical triangle setup are:
\[
\psi = A - 90^\circ, \quad \theta = 90^\circ - b, \quad \psi' = 90^\circ - B, \quad \theta' = 90^\circ - a, \quad x = c.
\]

Would you like further details or explanations on these derivations?

Here are some questions you might want to ask next:
1. Can you explain how to derive $\psi$ and $\theta$ step-by-step?
2. How do these spherical angles help in solving real-world problems?
3. What are the common applications of spherical trigonometry in astronomy?
4. Can you show an example problem involving spherical triangle calculations?
5. What is the significance of these angles and coordinates in navigation?

**Tip:** When working with spherical triangles, it's helpful to visualize the relationships between angles and sides on the sphere to better understand their geometric properties.

Explore the concept of angular distance $ \theta $ in spherical trigonometry. This page provides a detailed explanation on how $ \theta $ represents the angular distance between the XY-plane and the line connecting the origin $ O $ to point $ C $. Key formulas and their derivations are covered.

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