Plane trigonometry, by S.L. Loney.

88

TRIGONOMETRY.

The angle
RPN= 90~ - PNR = z RNO = OQ = A.
HMP MR + RP
Hence sin (A + B) = sin A OP= Op = M P
QN RP QN ON RPTNP
+ +
= O OP = ON OP NP OP
= sin A cos B + cos RPN sin B.. sin (A + B) = sin A cos B + cos A sin B.
Again cos (A + B) = cos A OP = O =O - MQ
OQ RN OQ ON RN VP
OP OP ON OP NP OP
= cos A cos B - sin RPN sin B..cos (A + B) = cos A cos B - sin A sin B.
89. The figures in the last article have been drawn only for the case
in which A and B are acute angles.
The same proof will be found to apply to angles of any size, due
attention being paid to the signs of the quantities involved.
The results may however be shewn to be true of all angles, without
drawing any more figures, as follows.
Let A and B be acute angles, so that, by Art. 88, we know that the
theorem is true for A and B.
Let A1=90~+A, so that, by Art. 70, we have
sin A = cos A, and cos A 1= - sin A.
Then sin (A + B)=sin {90~+ (A + B)} =cos (A +B), by Art. 70,
=cos A cos B - sinA sin B =sin A cos B + cos A1 sinB.
Also cos (A1 + B) = cos [90 + (A + B)] = - sin (A + B)
= - sin A cos B - cos A sin B = cos Al cos B - sin A1 sin B.
Similarly, we may proceed if B be increased by 90~.