Plane trigonometry, by S.L. Loney.

PRODUCT FORMULAE.

95

ON OM ONT- OM
Finally cos D - cos C= O -O = OR
oR oQ OR
MN_ SR 2SR PR
=OR ~ O P-R OR
= 2 sin SPR. sin POR. C+D. C-D
= 2 sin sin -
2 2
96. The student is strongly urged to make himself
perfectly familiar with the formulae of the last article and
to carefully practise himself in their application; perfect
familiarity with these formulae will considerably facilitate
his further progress.
The formulae are very useful because they change
sums and differences of certain quantities into products of
certain other quantities, and products of quantities are, as
the student probably knows from Algebra, easily dealt
with by the help of logarithms.
We subjoin a few examples of their use.
60 + 40 60 - 40
Ex. 1. sin 60 + sin 40 = 2 sin --- cos -- = =2sin 50 cos 0.
2 2
Ex. 2. cos-cos 70=2 sin 3070 sin7-30 =2 sin 50 sin 20.
2 2
75~ +15~. 75 - 15
sin 75 - sin 15~ 2 si 2
Ex. 3.. os 75~ + cos 15~ 750+ 15~ 750 -15~
2 cos 2 cos2 cos 45~ sin 30~ 1 -/3
=2 cos 450cos 30= tan 30 =3 57735......
[This is an example of the simplification given by these formulae; it
would be a very long and tiresome process to look out from the tables the
values of sin 75~, sin 15~, cos 750, and cos 15~, and then to perform the
division of one long decimal fraction by another.]