Some version of the following formula is often quoted for use in solving the "first geodetic problem" (aka "direct" or "forward" geodetic problem) on a spherical Earth:
$$ \phi_2 = \arcsin \left( \sin \left( \phi_1 \right) \cos \left( \delta \right) + \cos \left( \phi_1 \right) \sin \left( \delta \right) \cos \left( \theta \right) \right) $$
$$ \lambda_2 = \lambda_1 + \arctan \left( \frac { \sin \left( \theta \right) \sin \left( \delta \right) \cos \left( \phi_1 \right) } { \cos \left( \delta \right) - \sin \left( \phi_1 \right) \sin \left( \phi_2 \right) } \right) $$
Where:
- $\phi_1$ is the start latitude
- $\lambda_1$ is the start longitude
- $\delta$ is the central angle between points, computed from the desired distance of travel divided by Earth radius
- $\theta$ is the desired initial/forward azimuth of travel
- $\phi_2$ is the final latitude
- $\lambda_2$ is the final longitude
For example, this formula is shown on this popular collection of formulas, but no reference is given. A Wikipedia article on Great-circle navigation mentions that the formula can be derived using the formulas and identities of spherical trigonometry, and provides an expression for the central angle between two points, but a full derivation isn't provided.
I imagine this is just a matter of finding the right sequence of algebraic manipulations, but is there a straightforward derivation for the above formula from these spherical trigonometry and geodesy fundamentals?
Edit: I think the trick here is to reframe the problem as solving the spherical triangle between the start point, the North Pole, and the end point, with the azimuth being definitionally the interior angle between the start point and the North Pole.