🚀 go-pugleaf

Remember that exp(ix) = cos(x) + i sin(x)

so exp(ix)^n = exp(inx)

   (cos(x) + i sin(x))^n = cos(nx) + i sin(nx)

Considering real and imaginary parts

cos(nx) = REAL((cos(x)+i sin(x))^n)

sin(nx) = IMAG((cos(x)+i sin(x))^n)

and remember the binomial theorem.......

Rest left as an exercise!

==John ff

davidd@cnwl.igs.net (David Dixon) wrote:

>John Fitch <jpff@maths.bath.ac.uk> wrote:

>Yes.  I gave the same references.  Also pointed to Euler and DeMoivre
>as sources to review.  Awfully clever people those.  I liked reading
>about Euler; a very gentle person.

>David.

WOW!  My replies were cross-posted to your replies, although I did not
intend that.  I need to know more about how this works so that I can
control it better.

John Fitch <jpff@maths.bath.ac.uk> wrote:

Yes.  I gave the same references.  Also pointed to Euler and DeMoivre
as sources to review.  Awfully clever people those.  I liked reading
about Euler; a very gentle person.

David.

In an issue of Quantum magazine (some years back) there was a very 
interesting article about multiple angle formulas (not in simplest form) 
and their relationship to Pascal's Triangle as a source of their 
coefficients.  Unfortunately I don't have the publication date here at 
hand but I would be glad to find that information for people who want to 
look for the article. E-mail me privately if you are unable to find the 
article on your own.  My suspicion is that it was in 92, 93 or 94 at the 
latest.  I am not ceertain at this moment who the author of the piece 
was, although it may well have been George Berzsenyi who authors many 
articles and columns for that fine publication. (available from 
Springer-Verlag)

	See ya,
	Susan

-- 
	Susan Schwartz Wildstrom
	Walt Whitman High School
	Bethesda, MD

	In article <01bb9bf3$49e01100$1b5c0fcb@default>,
		Al <joiced@omen.com.au> wrote:


>cos 4x = 8cos^4 x - 8cos^2 x + 1)
>
>As the values of n increase, so too does the level of complexity of its
>expansion.
>
>I' ve established certain patterns and rules for the expansion in terms of 
>sin nx or cos nx. But am at a loss to find a diffinitive link (or rule)
>between the two. 


	Go to the library and look up "Chebyshev ploynomials".  Cos(nx) can
be written as an n-th degree polynomial in cos(x).  cos(nx)=P(cos(x)).  The
polynomial P is the n-th "Chebyshev polynomial".  You might find useful
stuff in books on "orthogonal polynomials".


	Mike Hardy

Michael Hardy
hardy@stat.umn.edu

I looked it up. It'sMay-June, 1993 Quantum on p. 48
	Susan

-- 
	Susan Schwartz Wildstrom
	Walt Whitman High School
	Bethesda, MD

Al (joiced@omen.com.au) wrote:

: Would anyone please be able to shed some light on the above problem for me,
: as i have reached a mental block on finding a totally perfect pattern or
: rule to decide the expansion from simply sin nx or cos nx.

Hmmm... okay... but I just hope this isn't for homework, or something.

sin(nx) = n[cos(x)^(n-1)]*x*sin(x) - nC3(cos(x)^(n-3))*sin(x)^3 + 
nC5[cos(x)^(n-5)]*sin(x)^5 + ... 

for n = 2, 3, 4, ...
and nCr = n!/[(r!)(n-r)!]

cos(nx) = (cos(x))^n - nC2[cos(x)^(n-2)]*sin(x)^2 + 
nC4[cos(x)^(n-4)]*sin(x)^4 - ...

with n and nCr being the same as stated above.

-- 
Jim Nastos
Mathematics '00
University of Waterloo

In article <01bb9bf3$49e01100$1b5c0fcb@default>, "Al" <joiced@omen.com.au> says:
>
>Could anyone one please help me with this investigation into multiple angle
>formulae i'm doing?  Any assistance would be greatly apprechiated.
>
>The information provided to me was:
>
>;sin 2x = 2.sin x.cos x
>          = sin x (cos x)
>
>;sin 3x = sin x (4cos^2 x -1)
>
>and,     cos 2x = cos^2 x - sin ^2x
>                      = 2. cos^2 x - 1
>                    
>       ;   cos 3x = 4.cos^3 x - 3.cos x 
>
>
>By using an extremly tedious method of expansion, re-arranging and
>substitution i managed to re-write further positive integer values of n for
>sin nx and cos nx.
>
>For example,
>  
>sin 4x = sin x (8cos^3 x - 4cos x)
>
>cos 4x = 8cos^4 x - 8cos^2 x + 1)
>
>As the values of n increase, so too does the level of complexity of its
>expansion.
>
>I' ve established certain patterns and rules for the expansion in terms of 
>sin nx or cos nx. But am at a loss to find a diffinitive link (or rule)
>between
>the two. 
>
>I have for example discovered that the rule to determine the first term of
>the expansion is ;   2^n-1 = first term. ie. 2^(4-1) = 8     (as above)
>
>Would anyone please be able to shed some light on the above problem for me,
>as i have reached a mental block on finding a totally perfect pattern or
>rule to decide the expansion from simply sin nx or cos nx.
>
>
>T.I.A,    Al         


Its a hell of alot easier to find both sin(nx) and cos(nx) by expanding
out cis(nx)=(cos(x)+i sin(x))^n and taking the real and imaginary parts
to find that cos(x)=Real(cis(x)^n) and sin(x)=Imag(cis(x)^n)