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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

849.0. "sin(nx) = f(sin(x),cos(x)) ???" by VIDEO::OSMAN (type video::user$7:[osman]eric.vt240) Thu Mar 31 1988 19:52

In a recent topic, Mr. Gilbert pointed out the following:

	sin 2x = 2 sin x cos x
	sin 3x = 3 sin x - 4 (sin x)^3
	sin 4x = cos x ( 4 sin x - 8 (sin x)^3 )
	sin 5x = 16 (sin x)^5 - 20 (sin x)^3 + 5 sin x


Can anyone figure out the pattern here and state the closed form for:

	sin (nx) = f(sin(x),cos(x))

Alternately, how about

	sin(nx) = f(sin((n-1)x))       [recursive formula]



/Eric

T.R	Title	User	Personal Name	Date	Lines
849.1	How about RTFB	MEIS::WOLFF	I feel the need, the need for speed	`Thu Mar 31 1988 19:58`	7
	I think this is a case for looking into a math formula collection, like "Handbuch der Mathematik". I look it up, and see if I can find something. Julian.
849.2	:-)	8BALL::HALLYB	You have the right to remain silent.	`Thu Mar 31 1988 20:41`	7
	> I think this is a case for looking into a math formula collection, > like "Handbuch der Mathematik". I think Eric was looking for something more direct, like Gilbert showed, only without putting the bucket of water on the floor. Oops, excuse me. Wrong note.
849.3	If you really want ot play with these...	AKQJ10::YARBROUGH	Why is computing so labor intensive?	`Thu Mar 31 1988 20:44`	6
	MAPLE handles this kind of expression quite well for fixed n; try > expand(sin(10*X), X); for example. It may give some insights into the general form, which MAPLE is able to compute quite quickly.
849.4	Here you have it...	AKQJ10::YARBROUGH	Why is computing so labor intensive?	`Thu Mar 31 1988 20:52`	8
	Aha, found it in Bronshtein & Semendyayev. Spoiler follows the <FF>. Sin(nx) = ncos(x)^(n-1)sin(x) - C(n,3)cos(x)^(n-3)sin(x)^3 + C(n,5)cos(x)^(n-5)sin(x)^5 - ... Cos(nx) = cos(x)^n - C(n,2)cos(x)^(n-2)sin(x)^2 + C(n,4)cos(x)^(n-4)sin(x)^4 - ...
849.5	Yupe, that's the book I meant.	MEIS::WOLFF	I feel the need, the need for speed	`Fri Apr 01 1988 00:19`	14
	Lynn, same book I looked in... but after I typed it in now I might as well post it. sin(nx) = n-1 /n\ n-3 3 /n\ n-5 5 n cos (x) sin(x) - ( ) cos (x)sin (x) + ( ) cos (x)sin (x) -... \3/ \5/ Julian.
849.6		CADM::ROTH	If you plant ice you'll harvest wind	`Mon Apr 04 1988 16:00`	4
	Such identities are much more transparent in terms of the complex exponential, exp(nit) = cos(nt) + isin(n*t). - Jim