Plane trigonometry, by S.L. Loney.

90

TRIGONOMETRY.

OfON OQ~+QM OQ NR
Also cos(A-B)=0 - = - - o = - +o
OP OP OP OP
OQ ON NR NP
O O O-+ P 0- = cos A cos B + sin NPR sin B,
ONT( OP, NP OP
so that cos (A - B) = cos A cos B + sin A sin B.
91. The proofs of the previous article will be found to apply to
angles of any size, provided that due attention be paid to the signs of
the quantities involved.
Assuming the truth of the formulae for acute angles, we can shew
them to be true universally without drawing any more figures.
For, putting A O=90~+A, we have,
(since sin Al= cos A, and cos A= -sin A),
sin (A1 - B) = sin [90~ + (A - B)] = cos (A - B) (Art. 70)
= cos A cos B + sin A sin B
= sin A1 cos B - cos Al sin B.
Also cos (Al - B) = cos [90+ (A - B)]= - sin (A - B) (Art. 70)
= - sin A cos B + cos A sin B
= cos A cos B + sin A1 sin B.
Similarly we may proceed if B be increased by 90~.
Hence the theorem is true for all angles which are not greater than
two right angles.
So, by putting A2= 90 ~+A,, we may shew the theorems to be true for
all angles less than three right angles, and so on.
Hence, by proceeding in this manner, we may shew that the theorems
are true for all angles whatever.
92. The theorems of Arts. 88 and 90 which give
respectively the trigonometrical functions of the sum and
differences of two angles in terms of the functions of the
angles themselves are often called the Addition and Subtraction Theorems.