arcs; and in order that these expressions may have the more generality, give to the radius any value R, instead of confining it to unity. This indeed may always be done in an expression, however complex, by merely rendering all the terms homogeneous; that is, by multiplying each term by such a power of R as shall make it of the same dimension, as the term in the equation which has the highest dimension. Thus, the expression for a triple arc

sin. 3A 3 sin. A 4sin'. A (radius = 1)
becomes when radius is assumed

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 a be supposed equal to B, we shall have

And, in like manner, by supposing B to become successively equal to 2A, 3A, 4A, &c, there will arise

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

COS2. A sin2. A

R

Cos. A. cos. 2A- sin. A. sin. 2A

cos. 2A =

cos. 3A =

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 cosine.

As

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

Add together the expanded expressions for sin. (в + A), sin. (BA), that is,

add

to

sin. (BA) sin. B . cos. A + cos. B. sin. A, sin. (BA) sin. B. COS. A cos. B . sin. A ; there results sin. (B+A)+ sin. (B — A) =2 cus. A. sin. B: whence, sin. (BA)=2 cos. A . sin. B · sin. (B — A). Thus again, by adding together the expressions for cos(B+A') and cos. (BA), we have

COS. (B+4)+cos. (B—A) = 2 cos. A COS. B;

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

sin. 2A 2 cos. A sin. A.

sin. 3A

sin. 4A

2 cos. A. sin. 2A

2 cos. A

sin. 3A sin. 2A.

(v.)

sin. na = 2 cos. A. sin. (~—1)▲ — sin. (~ — 2) a .

Cos. 2A= 2 cos. A. cos. A cos. 0 (1).

Cos. 3A 2 cos. A. cos. 2A-cos. A.

cos. (n-2)A..

(.xi.)

cos. 4A 2 cos. A. cos. 3A-cos. 2A. cos. NA=2 cos. A. cos. (n-1) A Several other expressions for the sines and cosines of multiple arcs, might readily be found: but the above are the most useful and commodious.

* 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.

20.

20. From the equation sin. 2a =

· 2 sin A. cos A

R

easy, when the sine of an arc is known, to half. For, substituting for cos A its value 2 sin A (R2-sin2 A). ✔✅✅ a)

there will arise sin 2A =

R

it will be

find that of its

√(R2 — sin2 A), This squared

gives R2 sin2 2A = 4R2 sin2 A 4 sin A.

A

Here taking sin a for the unknown quantity, we have a quadratic equation, which solved after the usual manner, gives

sin A = ± √ R2 ± 4R√ R*—sinˆ 2a.

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

sin AVR RVR sin' A'

or sin A√2R2R COS A:

}

(XII.)

by putting cos a' for its value ✔ R2- sin2 a′, multiplying the quantities under the radical by 4, and dividing the whole second 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

R

; whence,

sin ▲ . cos B = R 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)- R. Sin (A-B).. (XIV.) When A = B, both equa. xIII and xiv, become

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

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

COS2 AR. cos 2A + R2 (XVII.)

...

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

sin A.sin B = R. Cos (A - E) — R. cOS (A+B) (XVIII.) When AB, this formula becomes

sin2 A = R2 — R. cos 2A. . . (XIX.)

24. Mula

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

sin (A+B). Sin (A-B) = sin2 A sin2 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) × (sin Asin B), that is, to the rectangle of the sum and difference of the sines; it follows, that the first of these equa tions converted into an analogy, becomes

sin (4-B): Sin A-sin B :: sin ▲ + 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 ▲ : sin (n+1)a− sin na :: sin (n+1)a + sin na : sin (2n+1)a. ... (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 + B = A', and A-B = B'; then the half sum and the half difference of these equations will give respectively a=(A+B), and B=(A-B'). Putting these values of A and B, 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 formulæ :

sin (A+B). cos

sin (A-B'). Cos cos (A+B). cOS sin (A+B). sin Dividing the second of

be had

sin (A-B'). cos (A'+ B') sin (A+B). cos (A-B')

(A'— B') =

R(Sin A'+ sin B ́),

tan

,

factors of the first member of this equation, are

tan(A-B, and R

R

and tan(+) respectively; so that the equation

t

(XXII.)

This equation is readily converted into a very useful pre portion, 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.

VOL. III.

F

26. Operat

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 (A+B). tan ‡(A' — B') COS B'.

R2

=

'-COS A'

COS A+COS B

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

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 (AB), there results

then dividing both numerator and denominator of the second

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

sin

tan

COS

R

We و

28. We might now proceed to deduce expressions for the tangents, cotangents, secants, &c, of multiple arcs, as well as

some