FUNCTIONS OF SUMS AND SUMS OF FUNCTIONS

87. Functions of u + v in terms of the functions of u and v.

be acute angles. Draw P'D the radius OP(=1), and DC, P'B' 1 OA1, and ED || OA1, whence EP'D=u.

The radius being 1, P'D = sin v, and OD=

cos v; and

Use. These formulas are used in finding the sine and cosine of the sum of two angles when the sines and cosines of the angles are known.

88. Generality of [22] and [23]. In deducing formulas [22] and [23], the angles u, v, and u+v were taken, for convenience, less than 90°; we now purpose to show that these important formulas are true for all values of the angles u and v.

I. When u and v are acute, and u + v > 90°.

u and 90° v, and their sum, 180° — (u + v), are

Hence [22] and [23] are true for these angles. That is,

sin [180°—(u+v)]=sin (90° — u)cos(90° —v)+cos(90° — u)sin(90° — v), cos [180° — (u+v)] = cos(90° —u)cos(90° —v)+(sin 90° —u)sin(90°—v). Reducing these by Arts. 82, 85,

sin (u + v) = sin u cos v + cos u sin v,

cos (u + v) = cos u cos v sin u sin v.

Therefore [22] and [23] are true when u and v are any two acute angles.

II. When u and v are any positive angles.

Let u and v be any angles for which [22] and [23] are true. ̧

Let

90° + u = u', then 90° + u + v = u' + v.

Taking (1) the sines and then (2) the cosines of both members of these equations, whatever may be the value of u (Art. 82), we have

Substituting these values for cos u, cos (u+v), sin u, sin (u + v) in [23] and [22], we have, respectively,

sin (u' + v)= sin u' cos v + cos u' sin v,

cos (u' + v) = cos u' cos v

sin u' sin v.

Therefore, formulas [22] and [23] are true when u or v is increased by 90°, and also true for each repeated increase of one or the other angle by 90°. Hence, since they have been proved true for any two acute angles, they are true for any two positive angles whatever.

In a similar manner it may be shown that [23] is true when u and

The case in which the two angles u and v have contrary signs will be considered in Art. 89.

The other functions of u+v may be readily derived from [22] and [23].

Dividing [22] and [23] by cos u cos v, we have, (ƒ) Art. 21,

tan-11 = 45° = 1π.

89. Functions of u v in terms of the functions of u and v. Let be any angle less than 360°, then 360° v is a positive angle, and we may therefore substitute it for v in [22], which gives

sin (360° + u — v) = sin u cos (360° — v) + cos u sin (360° — v).

Now the angles 360° + u — v and u v have the same terminal, and so have 360°-v and v; therefore, sin (360°+u-v)=sin (u-v) and sin (360°-v)=sin- v =—sin v, cos (360° —v) = cos — v = cos v.

sin (uv) sin u cos v + cos u sin - v,

which shows that [26] may be derived from [22] by substituting v for v, and this proves the generality of [22] and also of [23], since [23] and [22] are interdependent.

Therefore, we may deduce the values of cos (u — v), tan (u — v), cot (uv) from [23], [24], [25] by substituting - v for v, which gives (Art. 81)

90. Functions of 2 u in terms of the functions of u.

Making v = u in [22], we have

sin (u+u) = sin u cos u + cos u sin u,

By these formulas, [30] to [33], the functions of twice an angle are found when the functions of the angle are known.

91. Functions of half an angle in terms of the functions of the angle. Take the formulas

Putting A in place of 2 u, and therefore A in

2 cos2 A = 1 + cos A (c), and 2 sin21⁄2 A

[34], sin AV

=

cos A 2

The sign before the radical, in each case, is to be determined by

Examples. By Art. 29, sin 30°, cos 30° = √ √3,

92. Sum and difference of functions in terms of the product of functions. [22]+[26], sin (u + v) + sin (u — v) = 2 sin u cos v

[22]-[26], sin (u + v) — sin (u — v) = 2 cos u sin v [23]+[27], cos (u + v) + cos (u — v) =2 cos u cos v [23]-[27], cos (u+v)

Making u+v = A and u — v = = B, and therefore u =

(a)

(b)

(c)

cos (u — v):

==

2 sin u sin v

and v = (A – B),

1

From (a), sin A + sin B =

2 sin (A+B) cos

(A – B)

[41]