🚀 go-pugleaf

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)