I know the spherical law of cosines, namely $\cos (c)=\cos (a) \cos (b)+\sin (a) \sin (b) \cos (C)$, where the lengths of each side arch of the spherical triangles are $a,b,c$ and the angle of the corner opposite to side $c$ is $C$.
Now consider Beijing ($39.9042°$N, $116.4074°$E), and Philadelphia ($39.9526°$N, $75.1652$°W) and that the circumference of the earth is $24,901$ miles.
I need to find the distance between the two cities, but I don't know how to proceed since I don't know any side lengths of a triangles (I also don't know which triangles I would use, I am assuming we then connect each city to the point ($0°$,$0°$). But if we did, I still don't know how to find those distances.
Any help would be appreciated to proceed with this problems. I have never worked with longitudes/latitudes.
Best Answer
Create a triangle with corners at your two cities (A and B) and the north pole (C). Then everything you need on the right side of your law can be read off from the lat/long of the cities. More specifically, $a$ and $b$ are $90^\circ$ minus each latitude, and $C$ is the difference in longitude.