(32.) Sines and cosines of multiple Arcs. sin 3x=3 sin x - 4 (sin x)3. sin 5x5 sin x-20 (sin x)3 + 16 (sin x)". sin 7x=7 sin x-56 (sin x)3 + 112 (sin a) - 64 (sin x). (sin x)' -- &c.} sin 4x= m (m2 - 12) (m2 - 32) 1.2.3.4.5 sin x. 2 cos x. - sin x 4 cos x 8 (cos a)3}. sin 6x= sin x {6 cos x-32 (cos x)3 + 32 (cos x)3 }. &c. = &c. m-2 sin mx=(-1) 2 sin x + sin 3x sin 5x= sin 7a &c. m (m _ 2%) (m — 4) 1...5 - sin x {1-4 (cos x)}. &c. sin a {1-12 (cos x)2 + 16 (cos x)1}. sin a {1-24 (cos x) +80 (cos x) - 64 (cos x)}. cos 6x=-1+18 (cos x)2 - 48 (cos x)*+32 (cos x). cos 5x + 5 cos x-20 (cos x)3 + 16 (cos x)5. cos 7x=-7 cos x + 56 (cos x)3 - 112 (cos x)5 + 64 (cos a)7. cos 2x=1-2 (sin x)2. cos 4x=1-8 (sin x)2 + 8 (sin x)*. cos 6x=1-18 (sin x)2 + 48 (sin x)a — 32 (sin x)o. cos 3x= cos x {1-4 (sin x)}. cos 5x = cos x {1− 12 (sin x)2 + 16 (sin x)*}. cos 7x= cos x {1-24 (sin x)2 + 80 (sin a) — 64 (sin x)6}. If n is even, the number of terms is n+1, and the last term, 2(-1)". If n is odd, the number of terms is (n+1), and the last term, (-1)(-1)(2n cos x). {(2 cos a)" — 1 — α (n − 2) sin nx = sin x (2 cos x)n = 3 If n is even, the number of terms is n, and the last term, (-1)-1.n cos x. If n is odd, the number of terms is (n+1, and the last term, (−1)}~− 1) ̧ -2 2 cos na = ( − 1)}"{(2 sin a)” — n (2 sin a)" – 2 r=1n+1 if n is even; r= (n + 1) if n is odd. n m-2 m -1 m-1 (2 cos x) n − 2m+1. r=±n if n is even; r = 1(n + 1) if n is odd. 2 sin na=(-1)*(-1)) (2 sin x)" — n(2 sin x)” – 2 n is even in the first and second of these series, and odd in the third and fourth: the last term and number of terms may be determined as in the two first series. These series are true only when n is a positive integer. (L. 370-97; W. Ch. iii; C. 466-84; Lagr. Calc. des Fonc.Leç. 11.) (33.) Tangents of multiple arcs. If n is odd the numerator and denominator must each be continued to (n + 1) terms; if even, to n, and respectively. n+1 terms (34.) Powers of the sine and cosine of an arc. = (cos 6x6 cos 4x + 15 cos 2x -10). &c. { - 1 (sin a)" = (-1)) cos na-n cos (n − 2) a x -1 1.3.5..(n-1) 1.2.3...n 2o (sin x)=(sin 7x-7 sin 5x + 21 sin 3x-35 sin x). n 2′′ – 1 (sin a)" — (— 1) { sin na — n sin (n − 2) x -1 = ·(tan x)2m · 2m-1 Ś(-1)-1, (tan x)2m 2m-2 If n is odd, r=s=1(n+1): if n is even, r=n, and s=3n+1. |