# Proof of part of haversine formula?

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?

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)
|   /
|  /
| /
-----------------------------------------------
``````

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)
``````
• :Okay.That's a good answer.Could you tell me how did you obtained the equation for chord AD as `2rsind &lambda/2`.Really from where did `2r` come? – justin Apr 16 '15 at 4:20
• @justin: Half the chord is r sin dλ/2 – Martin F Apr 16 '15 at 15:34