# Shortest Distance Between Two Latitude/Longitude Coordinates

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1. Jan 16, 2016

### transmini

1. The problem statement, all variables and given/known data

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.

2. Relevant 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: $cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)$

3. 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.

2. Jan 16, 2016

### SteamKing

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

3. Jan 16, 2016

### transmini

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.

4. Jan 16, 2016

### SteamKing

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

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

5. Jan 16, 2016

### vela

Staff Emeritus
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.