{m (sin a)* — &c.}: (32.) Sines and cosines of multiple Arcs. [1] Ascending series. sin 2x = cos 2.2 sin X. sin 4x=cOS X . {4 sin x -- 8 (sin x)"}. sin 6x=cosx. {6 sin x - 32 (sin x)' + 32 (sin x)"}. &c. = &c. m (m? — 2) sin ma = COS X m sin æ (sin x) 1.2.3 m (m? - 24) (m? 2) (m? – 4) + 1......5 sin 3x=3 sin x — 4 (sin x)". sin 5x=5 sin x – 20 (sin x)+ 16 (sin x)". sin 7x=7 sin x – 56 (sin x)3 + 112 (sin x)– 64 (sin x)? &c. =&c. m (m? – 14) sin mx=m sin a (sin x) 3 1.2.3 m (mé – 14) (m – 34) + (sin x)" — &c. 1.2.3.4.5 sin 2x=sin x. 2 cos ic. sin 40 = - sin x { 4 cos x 8 (cos a'); } sin 6x=sin {6 cos x {6 cos a -- 32 (cos x) + 32 (cos x)"}. &c. &c. m (m? — 22) (cos) 1.2.3 c 1...5 sin 3x= – sin x {1 – 4 (cos x)?\. sin 5 x = sin x {1 – 12 (cosa) + 16 (cos x)"}. sin 7x= -- sin « {1 -- 24 (cos x) + 80 (cos x)* -- 64 (cos x)}. &c. =&c, sin mæ=(-1)"7" sin « { m COS C + m? COS mw= (-1)3{ m-1 (m2 2 .m COS X sin mx=(-1)" sin a m*-1 1.2 (cos a)* – &c.} 1.2.3.4 cos 2x= -1+2 (cosa). cos 4x = +1–8 (cos x)2 + 8 (cos x)*. cos 6x= -1+18 (cos x) — 48 (cos x)* +32 (cos x). &c. &c. m2 (mo— 22) 1 - (cos x)2 + (cos x)*&c.} 1.2 1.2.3.4 cos 3x= - 3 cos x + 4 (cos x)". cos 5x= + 5 cos x — 20 (cos x)+ 16 (cosa)". cos 7x= -7 cos & + 56 (cos x)} – 112 (cos x)" + 64 (cos x)? &c. &c. - 1) cos mx=(-1) { (cos x)3 1.2.3 (m? — 1)(m? —- 3) (cos.2) – &c. 1.2.3.4.5 cos 2x=1-2 (sin x)?. cos 4x=1-8 (sin x)2 +8 (sin x)*. cos 6x=1-18 (sin x)* + 48 (sin x)4 – 32 (sin x). &c. &c. mo.(m* — 2) cos mx=1 1.2 (sin x)" 1.2.3.4 m2. (m? – 2)(m” – 4*) (sin x) + &c. 1....6 cos 3x=cos x {1 – 4 (sin x)"}. cos 5x = cos x {1-12 (sin x) + 16 (sin x)"}. cos 7x = cos x {1 {1-24 (sin x)2 + 80 (sin x)+ - 64 (sin x)"}. &c. &c. m_1 (m? – 1%)(m2–3) cos m x=cos x 31 -(sin x) + 1.2 (sin x)* – &c. 1.2.3.4 + mo &c.} [2] Descending series. n(n-3) 2 cos nx= (2 cos x)" — n (2 cos x)»–? + (2 cosa)" 1.2 n (n - 4)(n-5) (2 cos x)" – 6 + &c. 1.2.3 If n is even, the number of terms is In +1, and the last term, 2(-1)}. If n is odd, the number of terms is 1 (n + 1), and the last term, (-1)}(n-1)(2n cos w). (n-2) sin nx = sin a (2 1– (2 cos x)" =3 (n-3)(n— 4) (2 cos x)"– 5 – &c. 1 2 If n is even, the number of terms is in, and the last term, (-1){n-1.n cos x. If n is odd, the number of terms is (n+1, and the last term, (-1)}(n − 1). &. B — 1)#*{(2 sin x)" — n (2 sin a)*- cos nx=( n(n-3) + (2 sin )*— 4 2 n(n - 4)(n-5) (2 sin x)" – 6 + &c. Y 2.3 n m 1 m-2 m+2. m n 2 cos næ=Nśm(-1)m-1 (2 cos x )* – 2m 1 prin n+1 if n is even ; r=}(n+1) if n is odd. ß sin næ=sin w. Śm ( – 1)m-1. (2 cos x)" – 2m +1. g r={n if n is even ; r={(n+1) if n is odd. v If n is even, - 1 ml m 1 n - m sin næ=sin x. Sn(– 1 ym + fn - 1 m-1 (2 sin x)" – 2m+1. + 2 (n-2) sin nx=(-1)]*.cos x (2 sin x)n-1{ (2 sin a)" – 3 1 (n-3)(n- 4) (2 sin ~)*-5 –&c.} 1 . 2 sin nx=(-1)3(n = 1){(2 sin x)" — n(2 sin x)" – – a* e n(n-3) n(n-4)(n-5) (2 sin c)" - 4 (2 sin x)" – 6 + &c. c.} 3 cos nx=(-1)} (x – 1).cos x --- (2 sin x)*-s + 2 2 n 2 + (n-3)(n – 4) &c 1 . 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.) 5 tan 3 – 10 (tan abo + (tan abo tan 5 x 1-10 (tan x) + 5 (tan x) n(n-1)...(n—4) (tang)?-&c. 1 2 3 2 5 - 3) (tan x)* - &c. 1 2 1 2 3 4 I . . If n is odd the numerator and denominator must each be continued to }(n + 1) terms ; if even, to ln, and in +1 terms respectively. (C. 494-9; W. Ch. iii.) a{ (34.) Powers of the sine and cosine of an arc. cos 4x 4 cos 2 x + 3. &c. 2n – 1 (sin x)"=(-1){ cos nx - n cos (n - - 2) n(n-1) cos (n— 4)» – &c.} + 2ż-- 1.3.5.. (n-1) 1 1.2.3...n &c. sin (n — 4)- &c. + 2 . + – ) w.} 1 n 2m-2 Šm(–1)n-1 (tan x) 2m-2 2m-2 =kn +1. |