Proof of part of haversine formula?

Question

I'm trying to study how to find the distance between two points if their latitudes and longitudes are given.

I have a question about the proof of part of the haversine formula given at Math Forum. It says that the length of chord AD, two points at the same latitude, lat1, on a sphere of radius 1, is

2*sin(dlon/2)*cos(lat1)

but I couldn't get how they obtained it. Could you help me?

Answer 1

The radius, r, of the small circle joining all points at latitude, φ is

r = R cos φ

where R is the radius of the sphere. That simplifies to

r = cos φ

if we assume a "unit sphere" (R = 1) for convenience.

--------------------- A/D
|        r       φ /
|                 /
|                /
|               /
|a             /
|x            /
|i           /
|s          /
|          / R
|         /
|        /
|       /
|      /
|     /
|    /    ("side" view)
|   /
|  /
| /
|/ φ           equatorial radius
-----------------------------------------------

The chord length of a straight line, AD, joining two points on the same latitude is

AD = 2 r sin dλ/2

where dλ is the difference in longitude of A and D. Thus

AD = 2 R cos φ sin dλ/2

or

AD = 2 cos φ sin dλ/2

if R = 1

A-----------------D
 \       |       /
  \      |      /
   \     |     /
    \    |    / r
     \   |   /
      \  dλ /
       \   /
        \ /

    ("top" view)