§ 28. FUNCTIONS OF THE DIFFERENCE OF TWO ANGLES.

In a unit circle (Fig. 30) let the angle AOB=x, COB=y; then the angle AOC=x-Y.

У,

In order to express sin (x − y) and cos (xy) in terms of the sines and cosines of x and draw CFL OA, CD 1 OB, DE 1 OA, DGFC prolonged; then CD= siny, OD=cos y, and the angle DCG the angle EDC=x. And, sin (x − y) = CF = DE — CG.

B

= sin x; hence, DE

hence, DE = sin x × OD= sin x cos y.

DE

OD

CG

CD

hence, CG = cos x × CD= cos x sin y.

Therefore, sin (x-y)=sin x cos y cos x sin y.

cos (x − y) = OF = OE+DG.

[8]

Again,

OE

= cos x; hence, OE = cos x × OD=cos x cos y.

OD

DG

sinx; hence, DG= sinx × CD= sin x sin y.

CD
Therefore, cos(x − y) = cos x cos y + sin x sin y.

[9]

In this proof, both x and y are assumed to be acute angles; but, whatever be the values of x and y, the same method of proof will always lead to Formulas [8] and [9], when due regard is paid to the algebraic signs.

The general application of these formulas may be at once shown by deducing them from the general formulas established in § 27, as follows:

It is obvious that (x-y)+y=x. If we apply Formulas [4] and [5] to (x − y)+y, then

sin {(x− y)+y} or sin x = sin (x − y) cos y + cos (x — y) sin y, cos {(x−y)+y} or cos x = cos(x − y) cos y — sin (x — y) sin y.

Multiply the first equation by cos y, the second by sin y,

sin x cos y = sin(xy) cos2y+cos(xy) sin y cos y,

y:

cos x sin y—— sin (x − y) sin3y + cos (x − y) sin y cos y ; whence, by subtraction,

sin x cos y — cos x sin y=sin (x − y) (sin3y + cos3y).

But

·sin2y+cos2y=1;

sin (x − y) = sin x cos y

therefore, by transposing, cos x sin y.

Again, if we multiply the first equation by sin y, the second equation by cosy, and add the results, we obtain, by reducing,

cos (x − y) = cos x cos y + sin x sin y.

Therefore, Formulas [8] and [9], like [4] and [5], from which they have been derived, are universally true.

From [8] and [9], by proceeding as in § 27, we obtain

Formulas [4]-[11] may be combined as follows:

sin (x+y)=sin x cos y±cos x sin

y,

§ 29. FUNCTIONS OF TWICE AN ANGLE.

If y=x, Formulas [4]-[7], become

By these formulas the functions of twice an angle are found when the functions of the angle are given.

These values, if ≈ is put for 2x, and hence for x, become

By these formulas the functions of half an angle may be computed when the cosine of the entire angle is given.

The proper sign to be placed before the root in each case depends on the quadrant in which the angle ≈ lies. (§ 20.) Let the student show from Formula [18] that

§ 31. SUMS AND DIFFERENCES OF FUNCTIONS.

From [4], [5], [8], and [9], by addition and subtraction:

cos (x+y) cos (xy)=-2 sin x sin y;

or, by making x+y=A,

and x-y=B,

cos A+ cos B = 2 cos(A+B) cos † (A — B).
cos A-cos B-2 sin (A+B) sin (A-B).
†

[22]

[23]

From [20] and [21], by division, we obtain

1. Find the value of sin (x+y) and cos (x+y), when sin x

5

3, cos x=, sin y
, sin y=1, cos y=}}.

2. Find sin (90°-y) and cos (90°-y) by making x=90° in Formulas [8] and [9].

Find, by Formulas [4]-[11], the first four functions of:

3. 90°+y.

8. 360° ―y.

13. y.

14. 45°-y.

15. 45°+y.

16. 30°+y.

7. 270° +y.

12. x 270°.

17. 60°-y.

18. Find sin 3x in terms of sin x.
19. Find cos 3x in terms of cos x.
20. Given tan x= = 1; find cos x.
21. Given cotx= √3; find sin x.

22. Given sin x=0.2; find sine and cos x.
23. Given cos x=0.5;
24. Given tan 45° = 1;
25. Given sin 30° = 0.5;

26. Prove that tan 18°

find cos 2x and tan 2x.
find the functions of 22° 30!
find the functions of 15°.
sin 33° + sin 3°
cos 33° + cos 3o

If A, B, C are the angles of a triangle, prove that: 34. sin A+ sin B+ sin C≈ 4 cos A cos 1⁄2 B cos 1⁄2 C. 35. cos A+cos B+cos C=1+4 sin A sin B sin C. 36. tan A+tan B+tan C=tan AX tan B × tan C. 37. cot 4+cot B+ cot 1⁄2 C = cot A × cot 1 B × cot 1 C.

Change to forms more convenient for logarithmic computation: