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Re: Re: The formula of Abraham Moivre


On 24 Jan 2010, at 11:39, Sjoerd C. de Vries wrote:

> Hi Arnold,
>
> ExpToTrig[PowerExpand[FullSimplify[(Cos[x] + I*Sin[x])^n]]]
>
> but this is actually only true for n integer OR x positive reals.


Actually, it's true for any n (so all those assumptions 
Element[n,Integers] various people have posted are not needed). For 
example:

FullSimplify[
 ComplexExpand[(Cos[x] + I*Sin[x])^(3/2),
  TargetFunctions -> {Re, Im, Abs}], -Pi < x <= Pi] // ExpToTrig

cos((3 x)/2)+I sin((3 x)/2)

Is quite correct as is

FullSimplify[
 ComplexExpand[(Cos[x] + I*Sin[x])^(I),
  TargetFunctions -> {Re, Im, Abs}], -Pi < x <= Pi] // ExpToTrig

cosh(x)-sinh(x)

In full generality:

FullSimplify[
 ComplexExpand[(Cos[x] + I*Sin[x])^n,
  TargetFunctions -> {Re, Im, Abs}], -Pi < x <= Pi] // ExpToTrig

cos(n x)+I sin(n x)


Andrzej Kozlowski
>
> Cheers -- Sjoed
>
> On Jan 23, 2:35 pm, Arnold <sender999s... at gmail.com> wrote:
>> How by means of Mathematica to transform (Cos [x] +I* Sin [x]) ^n in 
Cos[n*x] +I*Sin [n*x]?
>>
>> Thanks.
>
>



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