Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

(32.) Sines and cosines of multiple Arcs.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

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.}

[blocks in formation]

sin 4x=

m (m2 - 12) (m2 - 32)

1.2.3.4.5

sin x. 2 cos x.

[ocr errors]

- 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.

[ocr errors][merged small][merged small][merged small][merged small][merged small]

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)}.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

cos 6x=-1+18 (cos x)2 - 48 (cos x)*+32 (cos x).

[blocks in formation]

cos 5x + 5 cos x-20 (cos x)3 + 16 (cos x)5.

[ocr errors]

cos 7x=-7 cos x + 56 (cos x)3 - 112 (cos x)5 + 64 (cos a)7.

[blocks in formation]

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.

[blocks in formation]

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}.

[blocks in formation]
[blocks in formation]

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

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

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

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

r=1n+1 if n is even; r= (n + 1) if n is odd.

[ocr errors]

n

[ocr errors]

m-2

m

[ocr errors][merged small]

-1 m-1

(2 cos x) n − 2m+1.

[blocks in formation]

r=±n if n is even; r = 1(n + 1) if n is odd.

[merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

2 sin na=(-1)*(-1)) (2 sin x)" — n(2 sin x)” – 2

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][merged small][merged small][merged small][merged small][merged small][ocr errors]

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.

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][merged small]

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

[blocks in formation]

(34.) Powers of the sine and cosine of an arc.

[blocks in formation]
[ocr errors]

=

(cos 6x6 cos 4x + 15 cos 2x -10).

&c.

{

- 1 (sin a)" = (-1)) cos na-n cos (n − 2) a

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

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).

[blocks in formation]

n

2′′ – 1 (sin a)" — (— 1) { sin na — n sin (n − 2) x

-1

=

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

·(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.

« ΠροηγούμενηΣυνέχεια »