10 ANALYTICAL PLANE TRIGONOMETRY. sin. 3▲ = 3 sin. ▲—4 sin 3. ▲ (radius = 1) 3A becomes when radius is assumed R R2 sin. 3a — R2 3 sin. A R2 or sin. 3A 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)= sin. 4A 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. 3A cos. 3A 2 sin. A COS. 4A R sin. A. cos. 2a+ cos. a . sin. 2a. .. sin. 5A R 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 sin. a. cos. 3A+ cos. A. sin. 3a. R Cos. A cos. 3A- sin. A COS. A. COS. 4A R sin. A sin. 2A sin. 3A sin. 4A (VIII.) sin. 2A 251/R2 — s2 sin. 4a =(4s—8$3) /R2 — s3 sin. 5a 5s-2083+16s5 sin. 6a =(6s – 32s3+32s5)√✓R2-s2 &c. COS. DA 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. As sin. a=s (IX.) (X.) Cos. ANALYTICAL PLANE TRIGONOMETRY. 11 Cos. A COS. 2A Cos. 3A COS. 4A cos. 5A= 16c5 cos. 6A C 2ca 403 804 8c2+1 20c3 + bc 48c4+18c2 32c€ 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 sin. (BA to sin. B. cos. A cos. B Sin. A, sin. B. COS. A COS. B. sin. a there results sin. (BA) + sin. sin. (B (B—A) = 2 Cos. a. sin. B. whence, sin. (B+A) = 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 (XI.) 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. 2A sin. 3A2 cos. A. A COS. NA = 2 cos. A cos. na COS. 2A. cos. (n (xi.) Several other expressions for the sines and cosines of multiple arcs, might readily be found: but the above are the most useful and commodious. the sin. A 1) A sin.(n-2)A. cos. 0 (=1). COS. A. 2 sin A COS A R (R2-sin2A), 20. From the equation sin. 2A= it will be easy, when the sine of an arc is known, to find that of its half. For, substituting for cos. A its value 2 sin. A ✓✓ (R2 there will arise sin. 2a squared gives R2 sin2 2a=4r2 sin2 a—4 sina a Here taking sin. A for the unknown quantity, we have a quad sin2 A 2A= R (x.) * 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 This C 12 ANALYTICAL PLANE TRIGONOMETRY. ratic equation, which solved after the usual manner, gives 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 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 sin. (A+B) + sin. (A—B): A 2 sin. B. COS. A sin. (A+B) —— sin. (4 —E) Cos. a . 2a 22. In like manner, by adding together the primitive ex- Cos. (A+B) + cos. (AB)= 2 cos. A cos. A. COS. B=R. COS. (A+B)+R. Co§. (4—B) (XVI.) COS2 A=1R. cos. 2A+R2 (XVII.) COS. B ; whence, 23. Deducting cos. (A+B) from cos.. (AB), there will remain. 2 sin. A . sin. B R cos. (A—B) — Cos. (A+B) 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. 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. ANALYTICAL PLANE TRIGONOMETRY. A-sin. B), that is, to the rectangle of the sum and differ- sin. (A 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. (A+B). COS (AB 1 2 B′). COS (A A B = 1 2 13 R(sin. A sin. B'), sin. B R(sin. A R (COS. B Dividing the second of these formulæ by the first, there will COS. B sin. (A-B) cos. (A+B) (A—B′) sin. Asin. B COS. -R sin. (AB) cos. (A+B) Cos. R tan. (A-B) and R R tan. (4′+B')' respectively; so that the tan. (A-B') sin. Asin. B (XXII.) 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. P 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. sin. A sin. A (A+B). ces. 4+ cos. B cot. (A+B) 1 COS. A 1+cos. A tan. A =tan. (A'—B'). COS. B .COS. A' hand COS. A COS. B COS. B COS. A tan. (A-B') Making B=0, in one or other of these expressions, there re- 1 cot. A 1. tan. A cot2 A cot. A cot. A tan2 A 1-COS. A tan. A : 27. Once more, dividing the expression for sin. (A+B) by sin. (A±B) sin. A sin. tan. →→ R tan. (A+B) R or, lastly, tan. (A + B) R2 Also, since cot we shall have tan. R2 cot. (A+B) cot. (AB)= R (tan. A tan. B) i sin. Asin. B COS. B-COS.A sin. A sin. B COS.B-COS. A (xxii.) COS. (XXIII) R2 tan. A. tan. B (XXIV.) cot. B cot. A 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 sin. |