R2

R

Hence then, if consistently with this precept, R be placed for a denominator of the second member of each equation v (art. 12), and if ▲ be supposed equal to в, we shall have sin. A. cos. A sin. A cos.

sin. (A+A)=

That is, sin. 2A=

R

B

And, in like manner, by supposing в to become successively equal to 2a, 3a, 44, &c. there will arise

sin. A. cos. 2a+ cos. a . sin. 2a.

R

sin. a. cos. 3A+ cos. A. sin. 3a.

sin. 3A

sin. 4A

sin. a . cos. 4a+† cos. a

sin. 5A

R

R
4A+ cos. A sin. 4A

(VIII.)

And, by similar processes, the second of the equations just referred to, namely, that for cos. (A+B), will give successively, Cos2 A sing

COS. 2A

R

COS. A. Cos. 2A

cos. 2a— sın. a
R

sin. 2A

(IX.)

R

19. If, in the expressions for the successive multiples of the sines, the values of the several cosines in terms of the sines were substituted for them; and a like process were adopted with regard to the multiples of the cosines, other expressions would be obtained, in which the multiple sines would be expressed in terms of the radius and sine, and the multiple cosines in terms of the radius and cosines.

Other very convenient expressions for multiple arcs may be obtained thus:

Add together the expanded expressions for sin. (BA), sin. (B—A), that is,

add

to

there results sin. (BA) + whence, sin. (B+A) =

sin. B. cos. A cos. B Sin. A,
sin. B. COS. A COS. B. sin. a
A;
sin. (B
(B—A) = 2 Cos. a. sin. B.
2 cos. A sin. B-sın. (B

▲).

Thus again, by adding together the expressions for cos. (BA) and cos. (BA), we have

COS. (BA) + cos. (BA) = 2 cos. A. cos. B, whence, cos. (B+A) 2 cos. A . COS. B COS. (BA). Substituting in these expressions for the sine and of BA, the successive values A, 2A, 3A, &c. instead of 2A, 3A, &c. instead of в; the following series will be produced.

sin. 2A-2 cos. A. sin. A. sin. 3A2 cos. A.

sin. 2A

sin. A

(x.)

sin.

sin. (n

1) A sin.(n-2)A.

A

cos. 0 (=1).

COS. A.

2 cos. A

cos. 3A

COS. 2A.

(xi.)

cos. na

COS. NA = 2 cos. A

cos. (n
(n − 1) a—cos.(n—2)a.

Several other expressions for the sines and cosines of multiple arcs, might readily be found: but the above are the

easy, when the sine of an arc is known, to find that of its half. For, substituting for cos. A its value

squared gives R2 sin2 2a=4r2 sin2 a—4 sina a Here taking sin. A for the unknown quantity, we have a quad

* Here we have omitted the powers of R that were necessary to render all the terms homologous, merely that the expressions might be brought in upon the page; but they may easily be supplied, when needed, by the rule in art. 18.

ratic

ratic equation, which solved after the usual manner, gives

sin. A = ±R2±R/R2 —sin3 2a

If we make 2a = A', then will a = 14′ and consequently, the
last equation becomes

sin. A =±√√/R2±R/R2 — sin3

or sin. A = ± 1/2R2 ± 2r cos, a':

(XII.)

by putting cos. A for its value R-sin A' multiplying the
quantities under the radical by 4, and dividing the whole se-
cond number by 2. Both these expressions for the sine of
half an arc or angle will be of use to us as we proceed.

21. If the values of sin. (A+B) and sine (A-B), given by
equa. v, be added together, there will result

2 sin, A. COS. B

R

; whence,

sin. (A+B) + sin. (A—B):
sin. A. cos. BR sin. (A+B)+R sin. (A-B). (XIII.)
Also, taking sin. (A—B) from sin. (A+B) gives

2 sin. B. COS. A
R

; whence,

sin. B. COS. AR sin (A+B)—1R . Sin (A—B). . (XIV.)
When AB both equa. xIII and XIV, become

Cos. a . 2a
sin. a=1r sin. 2A.. (XV.)

22. In like manner, by adding together the primitive ex-
pressions for cos. (A+B), COS. (A-B), there will arise

2 cos. A

COS. B

Cos. (A+B) + cos. (AB)=

; whence,

R

C

cos. A. COS. B=R. COS. (A+B)+R. Co§. (4—B) (XVI.) And here, when A=B, recollecting that when the arc is nothing the cosine is equal to radius, we shall have

COS2 A=1R. cos. 2A+R2 (XVII.)

23. Deducting cos. (A+B) from cos.. (AB), there will remain.

cos. (A—B) — Cos. (A+B)

2 sin. A . sin. B

; whence,

R

sin. a. sin. B=R. COS. (AB)-R. cos. (A+B) (XVIII.) When AB, this formula becomes

sin2 ARR. cos. 2A... (XIX.)

24. Multiplying together the expressions for sin. (A + B) and sin. (AE), equa. v, and reducing, there results

B

sin. (A+B). sin. (A—B) — sin2 A—şin3 B.

And, in like manner, multiplying together the values of cos.
(A+B) and cos. (A-B), there is produced

COS. (A+B). cos. (A-B) COS2 A--COS2 B.

Here, since sin2 A-sin2 B, is equal to (sin. a+sin. B) X (sin.

A-sin. B), that is, to the rectangle of the sum and difference of the sines; it follows, that the first of these equations converted into an analogy, becomes

sin. (A B): sin. A sin. B :: sin. A + sin. B: sin. (A + B) (XX.) That is to say, the sine of the difference of any two arcs or angles, is to the difference of their sines, as the sum of those sines is to the sine of their sum.

If A and B be to each other as n+1 to n, then the preceding proportion will be converted into sin. a: sin. (n+1) s— sin. na :: sin. (n+1) a+ sin. na : sin. (2n+1) ▲ . . . . (XXI.)

These two proportions are highly useful in computing a table of sines; as will be shown in the practical examples at the end of this chapter.

25. Let us suppose A+BA, and A-B=B'; then the half sum and the half difference of these equations will give respectively A (A+B), and B (AB). Putting these values of ▲ and в, in the expressions of sin. A. COS. B, sin. B. cos. A, cos. A. cos. B, sin. A. sin. B, obtained in arts. 21, 22, 23, there would arise the following formula:

sin.

sin.

2

(A+B). COS (AB
1
B′) =
B′). COS
COS } (A+B′)

(A

COS. 1 (A+B). cos / A B sin. 1 (A+B). sin (A′ — B

Dividing the second of these formulæ by the first, there will

be had

sin. Asin. B

sin. (AB) cos. (A+B) cos. ¿(A'—B′) ̊ sin. 1(A'+B′) sin. tan. Cos.

and

R

-R

COS.
— it
sin. tan.

follows that the two factors of the first member of this equa

tion, are

tan. (A-B) and

R

equation manifestly becomes

(XXII.)

tan. (A-B')
tan. 1(A+B′)

sin. Asin. B

sin. Asin. B′

This equation is readily converted into a very useful proportion, viz. The sum of the sines of two arcs or angles, is to their difference, as the tangent of half the sum of those arcs or angles, is to the tangent of half their difference.

26. Operating with the third and fourth formulæ of the preceding article, as we have already done with the first and second, we shall obtain

tan.

$

Making B=0, in one or other of these expressions, there re

These theorems will find their application in some of the investigations of spherical trigonometry.

27. Once more, dividing the expression for sin. (A+B) by that for cos. (A+B), there results

sin. (A±B) sin. A

COS. B sin. B COS. A

COS. (AB) COS. A. COS, B — sin. a. sin. B

then dividing both numerator and denominator of the second

fraction by cos. A. cos. B, and recollecting that

sin.

tan.

COS.

R

28. We might now proceed to deduce expressions for the tangents, cotangents, secants, &c. of multiple arcs, as well as some of the usual formulæ of verification in the construction of tables, such as

i

sin.

.