Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 164

Fig. 5.2 Derivations of the
sine formula

5.1.3 Formulae for Spherical Trigonometry

The formulae for spherical trigonometry are defined as the formulae applied to
obtain the unknown parts based on the given elements (sides, angles) of a spherical
triangle.

Sine Formula

In the spherical triangle ABC illustrated in Fig. 5.2 , the length of the sphere radius is
unity and we have:

sin a
sin A ﾼ

sin b
sin B ﾼ

sin c
sin C ,

ð 5 : 3 Þ

which means that in any spherical triangle, the sines of the sides are proportional to
the sines of their opposite angles), proved as follows.
Figure 5.2 is the trihedron O-ABC, a line BD is drawn perpendicular to the plane
OAC through point B. Then, through point D draw DE and DF perpendicular to OA
and OC. Join BE and BF; one obtains four right-angled plane triangles OBE, OBF,
BDE, and BDF. Meanwhile:

∠ BOC ﾼ a

, ∠

AOC ﾼ b

, ∠

AOB ﾼ c

, ∠

BED ﾼ A

, ∠

BFD ﾼ C :

It is given that:

BE
OB
BD

sin c
sin C ﾼ

BE BF

BF ﾼ

BD , and

OB