CHAPTER XII.

and

TRANSFORMATION OF PRODUCTS AND SUMS.

Transformation of products into sums or differences.
128. In the last chapter we have proved that

sin A cos B+cos A sin B=sin (A+B),
sin A cos B-cos A sin B=sin (A – B).

By addition,

2 sin A cos B=sin (A+B) + sin (A – B) .......................(1);

by subtraction

2 cos A sin B=sin (A+B) − sin (A – B) .........(2). These formulæ enable us to express the product of a sine and cosine as the sum or difference of two sines.

and

Again, cos A cos B-sin A sin B=cos (A+B), cos A cos B+sin A sin B=cos (A – B).

By addition,

2 cos A cos B=cos (A+B)+cos (A - B).........(3);

by subtraction,

2 sin A sin B=cos (A - B) - cos (A+B) .........(4).

These formulæ enable us to express

(i) the product of two cosines as the sum of two cosines;
(ii) the product of two sines as the difference of two

cosines.

129. In each of the four formula of the previous article it should be noticed that on the left side we have any two angles A and B, and on the right side the sum and difference of these angles.

For practical purposes the following verbal statements of the results are more useful.

2 sin A cos B=sin (sum)+sin (difference);
2 cos A sin B=sin (sum) – sin (difference);
2 cos A cos B=cos (sum)+cos (difference);
2 sin A sin B=cos (difference) - cos (sum).

N.B. In the last of these formula, the difference precedes the sum.

Example 1. 2 sin 74 cos 44 = sin (sum)+sin (difference)

=sin 114 + sin 34.

Example 2. 2 cos 30 sin 60=sin (30+60) – sin (30 – 60)

Example 4. 2 sin 75° sin 15° = cos (75° -15°) - cos (75°+15°)

=cos 60° - cos 90°

0

1

2

130. After a little practice the student will be able to omit some of the steps and find the equivalent very rapidly.

and

Transformation of sums or differences into products.

131. Since sin (A+B)=sin A cos B+cos A sin B, sin (A – B)=sin A cos B-cos A sin B ;

by addition,

sin (A+B)+sin (A – B) = 2 sin A cos B .........(1); by subtraction,

sin (A+B) sin (A-B)=2 cos A sin B.........(2). Again, cos (A+B)= =cos A cos B-sin A sin B, and cos (A-B)=cos A cos B+sin A sin B.

By addition,

cos (A+B)+cos (A – B)=2 cos A cos B.......................(3):

by subtraction,

cos (A+B)-cos (A-B)=-2 sin A sin B

=2 sin A sin(- B)......(4).

By substituting for A and B in the formulæ (1), (2), (3), (4), we obtain

132. In practice, it is more convenient to quote the formulæ we have just obtained verbally as follows :

sum of two sines=2 sin (half-sum) cos (half-difference); difference of two sines=2 cos (half-sum) sin (half-difference); sum of two cosines=2 cos (half-sum) cos (half-difference); difference of two cosines

=2sin (half-sum) sin (half-difference reversed)

Example 4.

H. K. E. T.

cos 70° - cos 10°-2 sin 40° sin ( -30°)
= -2 sin 40° sin 30° - sin 40°.

8

133. The eight formulæ proved in this chapter are of the utmost importance and very little further progress can be made until they have been thoroughly learnt. In the first group, the transformation is from products to sums and differences; in the second group, there is the converse transformation from sums and differences to products.

Many examples admit of solution by applying either of these transformations, but it is absolutely necessary that the student should master all the formula and apply them with equal readiness.