Plane trigonometry, by S.L. Loney.

174 TRIGONOMETRY.
In the triangle ABD, we have
AD
A = sin B, so that AD =c sin B.
AB
In the triangle ACD, we have
AD = sin, so that AD = b sin C.
AC
[If the angle C be obtuse, as in the second figure, we have
AD
= sin ACD= sin (180~- C)=sin C (Art. 72),
so that AD= b sin C.]
Equating these two values of AD, we have
c sin B =b sin C,
sin B sin C
z.e.
b c
In a similar manner by drawing a perpendicular from
B upon CA we have
sin C sin A
c a
If one of the angles, C, be a right angle as in the third
figure we have sin C = 1,
a a b
sin A =, and sinB=-.
c C
sinA sinB I sinC
Hence - - -
a b c c
We therefore have, in all cases,
sinA sin B sin C
a b c