Here taking sin. A for the unknown quantity, we have a quadratic equation, which solved after the usual manner, gives sin. A±√R2 ± {R√✓ R2 - sin. 2A. If we make 2A = A', then will A = A'; and consequently the last equation becomes = sin. A±√R2±R/R2 — Sin2. 2a' } (XII.) by putting cos. A' for its value R2-sin.' 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 sin. (a — B), given by equa. v, be added together, there will result sin (A+B) + sin (AB) = 2 sin A. COS B sin A. COS BR. Sin (A+B) + R ; whence R sin (A—B). . (XIII.) Also, taking sin (AB) from sin (A + B), gives sin B. COSA = When AB, both R. Sin (A+B)—¦R. sin (A-B).. (XIV.) equa. xIII and xiv, become COS A. sin A = JR Sin 2A.. (XV.) 22. In like manner, by adding together the primitive expressions for cos (A + B), COS (AB), there will arise COS (A+B) + cos (A — B) = 2 cos A. COS B R os; whence, 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 COS (AB) — Cos (A + B) = 2 sin A. sin B ; whence, R sin A. sin B = R. COS (A-B)-R. COS (A+B) (XVIII.) When AB, this formula becomes 24. Multiplying together the expressions for sin (A + B) and sin (AB), equation v, and reducing, there results sin (A+B). sin (A — B) sin2 A-sin3 B. And, in like manner, multiplying together the values of cos (A+B) and cos (AB), there is produced (sin sin B), that is, to the rectangle of the sum and dif. ference of the sines; it follows, that the first of these equations converted into an analogy, becomes sin (1—B) : sin a-sin B :: sin B + 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 ton, then the preceding proportion will be converted into sin a: sin (n + 1) ▲ — 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 ▲ -BB'; 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 : Dividing the second of these formulæ by the first, there will factors of the first member of this equation, are tan (A-B') R R and and(+), respectively; so that the equation tan1⁄4(a'+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 ('B'). tan }(A—B′) COS B' -COS A 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 sin (A+B) sin A. COS B sin B. COS A sin A. Sin B COS (A+B) COS A. COS B then dividing both numerator and denominator of the second fraction, by cos A. cos B, and recollecting that sin tan (XXIII.) ' R2 tan A. tan B. tan (A+B) tan Atan B reduction, becomes cot (A + B) = cot A. cot BR2 cot B cot A ... (XXIV.) we 28. We might now, by making a = b, a = 2b, &c. proceed to deduce expressions for the tangents, cotangents, secants, &c. of multiple arcs; but we shall merely present a few for the tangents, as We might again from the obvious equation sec A-tan 2A=sec 2B-tan 3Â, deduce the expression sec Atan A sec B = tan B sec Btan B Sec A tan ai and so, for many other analogies. We might investigate also some of the usual formulæ of verification in the construction of tables, such as Bin (54°+) sin (54°-A)-sin (18o+A) —sin (18°—A) sin (90°~A) sin Asin (36o — ▲) + sin (72° + x) == sin (36o + a)+ sin (72o — 4). . &c. &c. But, as these inquiries would extend this chapter to too great a length, we shall pass them by; and merely investigate a few properties where more than two arcs or angles are concerned, and which may be of use in some subsequent parts of this volume. 29. Let A, B, C, be any three arcs or angles, and suppose radius to be unity; then sin (B+c)= sin A. sin csin B. Sin (A+B+c) sin (A+B) For, by equa. v, sin (A+B+c)=sin A. cos (B+c)+cos A sin (B+c), which, (putting cos B. Cos C-sin B. sin c for cos (B+c)), is-sin A COS B. cos c-sin A. sin B. sin ct COS A. Sin (B+c); and, multiplying by sin B, and adding sin A. sin c, there results sin A. sin c+sin B. sin (A+B+C) sin A. COS B. cos c. sin в+sin A. sin c. cos2 B+cos A. sin B. sin (B+C)=sin a COS B. (sin B cos c+cos в sin c) +cos A. Sin B. sin (B+c) = sin A COS B. sin (B+ c) + cos A. sin a. sin (B+c) : = (sin A COS B+COS A. sin B) X in (B+c) sin (A+B). sin (B+ c). Consequently, by dividing by sin (A+B), we obtain the expression above given. In a similar manner it may be shown, that sin (B-C)= sin ▲ . sin c—gin B . sin (A→B+C) sin (A-B) 30. If A, B, C, D, represent four arcs or angles, then writing c+D for c in the preceding investigation, there will result, sin (B+C+D)= sin a . sin (c†D)+sin B. Sin (A+B+C+D) ̧ sin (A+B) A like process for five arcs or angles will give sin A. sin(c+D+x)+sin B. Sin(A+B+C+D+B) sin (A-+-B) sin (B+C+D+E)= sin A. sin(c+D+....) +sin B. sin(A+B+C+...) sin (B+C+....L)= sin (c-A) sin c. cos a-sin a. cos c, sin (A-B)=sin A. COS B-sin B. COS A. Multiplying the first of these equations by sin A, the second by sin B, the third by sin c; then adding together the equations thus transformed, and reducing; there will result, sin A. sin (B-c)+sin B. sin (c-A)+ sin c. sin(A-B)=0, cos A. sin (B—c)+cos B. sin (c-A)+cos c. sin(a—b)=0. These two equations obtaining for any three angles whatever, apply evidently to the three angles of any triangle. 32. Let the series of arcs or angles A, B, C, D contemplated, then we have (art. 24), sin (A+B). sin (A-B) sin (B+c). sin (B-C) = sin2 A-sin2 B, sin B-sin2 C, sin (c+D). sin (c-D) = sin3 c-sin2 D, &c. &c. &c. sin (L+A). sin (L-A) = sin' L—sin3 A. L, be If all these equations be added together, the second member of the equation will vanish, and of consequence we shall have ... sin (A+B). sin (A—B)+sin (B+c). sin (B—c)+&c .. +sin (L+A)+sin (L-A)=0. Proceeding in a similar manner with sin (A-B), cos (A+B), sin (B-C), COS (B+ c), &c. there will at length be obtained COS (A+B). Sin (A-B)+COS (B+c). sin (B−c)+&c. . . +cos (L+A). sin (L—a)=0. ..... 33. If the arcs A, B, C, &c. . . L form an arithmetical progression, of which the first term is 0, the common diffeence D', and the last term L any number n of circumferences; then will B A=D', € — B=v', &c. A + B=2n', b+c=3d', &c.: and dividing the whole by sin D', the preceding equations will become sin D' + sin 3D + sin 5D+ &c. = 0; } (XXV.) cos D' + cos 3D' + cos 5D' + &c. = 0. If E' were equal 2D', these equations would become sin D'+sin (DE) +sin (D'+2E) +sin (D'+3E') + &c. = 0, COS D'+cos(D'+E')+cos(D'+2E')+cos. (D'+3E')+&c. =0. 34. The last equation, however, only shows the sums of sines and cosines of arcs or angles in arithmetical progression, when the common difference is to the first term in the ratio of 2 to 1. To investigate a general expression for an infinite series of this kind, let ssin a+sin (A+B) +sin (A+2B) +sin (A+3B) + &c. Then, since this series is a recurring series, whose scale of relation is 2 cos B-1, it will arise from the developement of a fraction whose denominator is 1-2z. cos B + 22, making z = 1. Now this fraction will be = sin A+≈ [sin (A + B) — 2 sin a . COS B) 1-22 .COS B +22 |