# Shortest Distance Between Two Latitude/Longitude Coordinates

## 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.

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

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
Staff Emeritus
Homework Helper

## 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.

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

vela
Staff Emeritus