Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 165

BF
OB
BD

sin a
sin A ﾼ

BE BF

BE ﾼ

BD :

OB

Hence:

sin a
sin A ﾼ

sin c
sin C :

Similar results hold for the other sides and angles:

sin a
sin A ﾼ

sin b
sin B :

Combining these two series gives the sine formula.

Other Formulae

Other commonly used formulae are provided below without derivations (see, e.g.,
Casey 2005).
The law of cosines for sides is:

cos a ﾼ cos b cos c þ sin b sin c cos A ð 5 : 4 Þ

The law of cosines for angles is:

cos A ﾼ cos B cos C þ sin B sin C cos a ð 5 : 5 Þ

The laws of sines and cosines is:

9
= sin a cos B ﾼ sin c cos b cos c sin b cos A

sin a cos C ﾼ sin b cos c cos b sin c cos A

ð 5 : 6 Þ

sin A cos b ﾼ sin C cos B þ cos C sin B cos a ;

sin A cos c ﾼ sin B cos C þ cos B sin C cos a

The laws of cotangents is:

9
=
;

cot a sin c ﾼ cos c cos B þ sin B cot A

cot a sin b ﾼ cos b cos C þ sin C cot A

ð 5 : 7 Þ

cot A sin C ﾼ cos C cos b þ sin b cot a

cot A sin B ﾼ cos B cos c þ sin c cot a