45. Formulæ for the Transformation of Sums into Prod. ucts. - From the four fundamental formula of Arts. 43 and 44 we have, by addition and subtraction, the following:

sin (a + y) + sin (x - y)= 2 sin a cosy. (1) sin (x + y) - sin (x - y)=2 cos x siny. (2) cos (x + y) + cos (x - y)= 2 cos x cos y . (3) cos (x - y) - cos (x + y)= 2 sin x siny.

.

.

.

=

These formulæ are useful in proving identities by transforming products into terms of first degree. They enable us, when read from right to left, to replace the product of a sine or a cosine into a sine or a cosine by half the sum or half the difference of two such ratios. Let x+y=A, and x — y=B.

.: x= {(A + B), and y=} (A - B). Substituting these values in the above formulæ, and putting, for the sake of uniformity of notation, a, y instead of A, B, we get

sin x + sin y=2 sin } (x + y) cos } (x - y).. (5) sin x -

- sin y= 2 cos } (x + y) sin } (x — y). . (6) cos x + cosy = 2 cos } (x + y) cos } (x - y). (7)

COS Y - cos x = 2 sin } (x + y) sin } (x - y). · (8) The formulæ are of great importance in mathematical investigations (especially in computations by logarithms); they enable us to express the sum or the difference of two sines or two cosines in the form of a product. The student is recommended to become familiar with them, and to com mit the following enunciations to memory: Of any two angles, the Sum of the sines = 2 sin sum . cos 4 diff.

2 } · Diff. 66

= 2 cos sum • sinfdiff.

Sum of the cosines = 2 cos 4 sum · cos } diff.
Diff. 66

= 2 sin sum. sin diff.

EXAMPLES.

1. sin 5 x cos3x =} (sin 8x + sin 2x).
For, sin 5x cos3x = } {sin (5x + 3x)+sin (5x – 3x)}

= 1 (sin 8 x + sin 2x).
2. Prove sin o sin 30 =} (cos 24 - cos 40).

20 3. 2 sin 0 cos •

= sin ( + 0) + sin (0 - 0). 4.

2 30

2 sin 20 cos 3 = sin(20+30) + sin (29-30). 5.

sin 60° +sin 30o=2 sin 45° cos 15°.

sin 40° - sin 10° '= 2 cos 25° sin 15°. 7. sin 100+sin 60= 2 sin 8 A cos 20.

(6

66

6.

66

46. Useful Formulæ. - The following formula, which

are of frequent use, may be deduced by taking the quotient of each pair of the formulæ (5) to (8) of Art. 45 as follows:

}} x
1.

sin x + siny _ 2 sin 1(x + y) cos(a - y)
sin x — sin y 2 cos(x+y) sin (2-y)

tan : (x + y) cot }(x - y)
tan} (x + y).

(Art. 24)
tan }(- y)
The following may be proved by the student in a similar

47. The Tangent of the Sum and Difference of Two Angles. - Expressions for the value of tan(x + y), tan (x - y), etc., may be established geometrically. It is simpler, however, to deduce them from the formulæ already established, as follows:

Dividing the first of the x, y' formulæ by the second, we have, by Art. 23,

sin (x + y)

sin x cos y + cos2 siny. tan (x + y)

cos (x + y) cos x cos y — sinx siny Dividing both terms of the fraction by cos x cos y,

=

(1)

tan x + tany

(Art. 23)

1- tan x tany In the same manner may be derived

tan x — tany tan (x – y)=

[+ tan x tany

cotx coty - 1 Also, cot (x + y)=

cotx + coty

cotx coty +1 and cot (x – y)

coty — cotx

(2)

=

(3)

(4)

EXERCISES.

Prove the following:

tan x + 1
1.
tan (x + 450) =

1 – tan a

5. sin (x + y) sin (x - y) = sinox – sinʼy

2 — =

= cos'y – cosax.

6. cos (a + y) cos (x - y) = = cos’ x — sinay

10. If tan x = I and tany= \, prove that tan (x + y)= 4, and tan (2 — y)= g.

11. Prove that tan 15° 2 – V3.

12. If tan x=, and tany=, prove that tan(x + y)=1. What is (x + y) in this case?

48. Formulæ for the Sum of Three or More Angles. — Let x, y, z be any three angles; we have by Art. 43,

sin (x + y +z)=sin (x + y) cosz + cos (x + y) sin z

= sin x cos y cos z + cos x sin y cos z

+ cos x cos y sinz – sin x siny sinz · · (1) In like manner, cos (x + y +x)= cos a cosy cos z sin x sin y cos z

- sin x cos y sin z
Y

cos x siny sinz. (2)

=

.

Dividing (1) by (2), and reducing by dividing both terms of the fraction by cos x cos y cos z, we get

tan x + tan y + tanz tan (x+y+z)=

tan x tan y tanz

(3) 1 – tan x tany – tany tanz - tanz tan x

EXAMPLES.

1. Prove that sin x + siny + sin % - sin (x + y +z)

=

= 4 sin }(x + y) sin }(y + z) sin }(z + x).

=

By (6) of Art. 44 we have

sin x — sin (x +y+)=–2cos }(2x + y +z) sin }(y+z), and

siny+sinz=2 sin }(y + 2) cos }(y – z). .. sin x + siny + sinz - sin (a +y+z)

=2 sint(y+z) cos }(y-2) – 2 cos }(2x+y+z) sin }(y+z) =2 sint(y+z) {cos }(y-2) - cos }(2x + y +z)} =2 sin }(y+z) 2 sin } (x + y) sin }(x + x) =4sin } (x + y) sin ž (y+z) sin }(z + x).

Prove the following:

2. cos x + cosy + cosz + cos (x + y + x)

= 4 cos }(y + 2) cos }(2 + x) cos }(x+y).