Shortest Distance Between Two Latitude/Longitude Coordinates

[transmini]

Homework Statement

We need to find the shortest distance between two given cities. For this I'll use Bangkok, Thailand (13°N, 100°E) and Havana, Cuba (23°N, 82°W ). Earth is assumed to be perfectly spherical with a radius of 6.4x10⁶m. These aren't the places we were given but the coordinates are similar.

Homework Equations

The only equations we have are
The Law of Sines: $\frac{sin(a)}{sin(A)} = \frac{sin(b)}{sin(B)} = \frac{sin(c)}{sin(C)}$
The Law of Cosines for Sides: $cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)$
and
The Law of Cosines for Angles: $cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)$

The Attempt at a Solution

Honestly I'm not even sure how to start with this. I began by drawing a spherical triangle and labeling the points, with two points being the coordinates of the cities and the third being at (0°, 0°). Continuing from here is where I get lost seeing as how I know nothing about math with spherical triangles aside from the equations given above. Once I find the angular length of the great circle arc connecting the two cities, I know that I use the relation $s = r\theta$ where s is the arc length, but I have no idea how to find that side when I only have two coordinates and no side lengths or angles.

 

[SteamKing]

Not a physics problem, let alone an advanced physics problem. Moved to Pre-calculus math HW forum.

 

[transmini]

Not a physics problem, let alone an advanced physics problem. Moved to Pre-calculus math HW forum.

Oops, my bad. I received it for an astrophysics class and saw a similar post in that forum so I assumed that's where it would be. Thanks for the info though.

 

[SteamKing]

Homework Statement

We need to find the shortest distance between two given cities. For this I'll use Bangkok, Thailand (13°N, 100°E) and Havana, Cuba (23°N, 82°W ). Earth is assumed to be perfectly spherical with a radius of 6.4x10⁶m. These aren't the places we were given but the coordinates are similar.

Homework Equations

The only equations we have are
The Law of Sines: $\frac{sin(a)}{sin(A)} = \frac{sin(b)}{sin(B)} = \frac{sin(c)}{sin(C)}$
The Law of Cosines for Sides: $cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)$
and
The Law of Cosines for Angles: $cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)$

The Attempt at a Solution

Honestly I'm not even sure how to start with this. I began by drawing a spherical triangle and labeling the points, with two points being the coordinates of the cities and the third being at (0°, 0°). Continuing from here is where I get lost seeing as how I know nothing about math with spherical triangles aside from the equations given above. Once I find the angular length of the great circle arc connecting the two cities, I know that I use the relation $s = r\theta$ where s is the arc length, but I have no idea how to find that side when I only have two coordinates and no side lengths or angles.

There's plenty of information on the web about spherical trig and such.

This article may help:

https://en.wikipedia.org/wiki/Great-circle_distance

 

[vela]

Forget about spherical trig for a moment. Just look at it as a vector problem, and you're trying to find the angle between two vectors. Start by figuring out the unit vectors that point in the direction from the center of the Earth to each city.