Question regarding the earth's spherical geometry

Question

Hello I have a question and it is as follows: What is a general formula to derive an angle from true north such that I know how to face a certain object assuming I know the longitude and latitude of my position and the longitude and latitude of the place? To clarify I will provide an example: If I was in Seattle, Washington, USA $(47.6062^\circ\text{ N}, 122.3321^\circ\text{ W})$ what angle from true north would I have to turn to face Moscow, Russia $(55.7558^\circ\text{ N}, 37.6173^\circ\text{ E})$ such that if I walk along this direction I will reach Moscow without having to turn left nor right.

Answer 1

The north pole $N$ and the two cities $S$, $M$ are the vertices of a spherical triangle. From the given data you can easily compute the angle $\alpha$ at $N$, as well as the lengths $b$, $c$ of the two sides $NS$, $NM$: $$\alpha=122.3^\circ+37.6^\circ,\quad b=90^\circ-47.6^\circ,\quad c=90^\circ-55.8^\circ\ .$$

Use the textbook formula $$\cos a=\cos b\>\cos c+\sin b\>\sin c\>\cos\alpha$$ to determine the length $a$ of $SM$ and then the spherical sine law $${\sin\gamma\over\sin c}={\sin\alpha\over\sin a},\qquad{\rm resp.},\qquad\sin\gamma={\sin c\over\sin a}\sin\alpha$$to determine the angle $\gamma$ at $S$.

Draw a figure!