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{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
m (m? - 2) (m— 4*)
+

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,

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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
sin æ {1 (cos x)2

1.2
(m? - 12) (m? — 3)

(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)? +

(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.}

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[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

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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

[blocks in formation]

n - m

sin næ=sin x. Sn(– 1 ym + fn - 1 m-1

(2 sin x)" – 2m+1.

[ocr errors]

+

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

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+

(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.)

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5 tan 3 – 10 (tan abo + (tan abo tan 5 x

1-10 (tan x) + 5 (tan x)
n(n-1)(n-2)

n(n-1)...(n—4)
ntana
(tan x)+

(tang)?-&c. 1 2 3

2

5
tannx
n(n-1)
n(n-1)(n-2)(n

- 3)
1
(tan x)2 +

(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.
2 (sin x)* = -(cos 2 x - 1).
2 (sin x)* =

cos 4x 4 cos 2 x + 3.
2 (sin x) (cos 6x — 6 cos 4x + 15 cos 2x – 10).
&c.

&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
2(sin x) = -(sin 3x - 3 sin x).
24 (sin ~)* = sin 5x – 5 sin 3x + 10 sin x.
2® (sin x) = -(sin 70-7 sin 5x + 21 sin 3x - 35 sin x).
&c.

&c.
2~-1(sin a)" = (– 197{sin na-n sin (n − 2) w
n(n − 1)

sin (n — 4)- &c.

+

2

.

+

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– ) w.}

1

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n

2m-2

Šm(–1)n-1 (tan x)
If n is odd, r=s=} (x + 1): if n is even, r=in, and

2m-2 2m-2

=kn +1.

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