Using similar arguments associated with the representation of cos αand cos β, one
can show
cos b= cos ccos a+ sin csin acos β
cos a= cos bcos c+ sin bsin ccos α
(9 .15)
The equations (9.14) and (9.15)) are known as the law of cosines for the spherical
triangle ABC.
Replace the dot product in equation (9.11) by a cross product and show
sin γ=|(ˆeB׈eC)×(ˆeA׈eC)|
|ˆeB׈eC||ˆeA׈eC|
(9 .16)
The cross product relation (6.30), repeated here as
(A×B)×(C×D) = C
[
D·(A×B)
]
−D
[
C·(A×B)
]
(9 .17)
can be used to simplify the numerator of equation (9.16). One can use properties of
the scalar triple product to write
(ˆeB׈eC)×(ˆeA׈eC) = ˆeA[ˆeC·(ˆeB׈eC)] −ˆeC[ˆeA·(ˆeB׈eC)]
=ˆeA[ˆeB·(ˆeC׈eC)] −ˆeC[ˆeA·(ˆeB׈eC)]
=−eˆC[ˆeA·(ˆeB׈eC)]
(9 .18)
so that
|(ˆeB×eˆC)×(ˆeA×eˆC)|=eˆA·(ˆeB׈eC)
The triple scalar product relation shows that
sin γsin asin b=|ˆeA·(ˆeB׈eC)|
sin αsin bsin c=|ˆeB·(ˆeC׈eA)|
sin βsin csin a=|ˆeC·(ˆeA׈eB)|
(9 .19)
and the scalar triple product relation implies that
sin αsin bsin c= sin βsin csin a= sin γsin asin b (9 .20)
Divide each term in equation (9.20) by sin asin bsin cto show
sin α
sin a=
sin β
sin b=
sin γ
sin c (9 .21)
which is known as the law of sines from spherical trigonometry.
