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Topic: Rational values of sine
Replies: 5   Last Post: May 25, 2012 6:16 PM

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 Daryl McCullough Posts: 2,871 Registered: 1/29/05
Re: Rational values of sine
Posted: May 25, 2012 5:48 PM

On Friday, May 25, 2012 4:31:45 AM UTC-4, Robin wrote:
> On 25/05/2012 02:32, Daryl McCullough wrote:
> > Is there a simple answer to this question: for which
> > rational numbers p (0<= p<= 1/2) is sin(p pi) rational?
> >
> > Off the top of my head, p can be 0, 1/6, 1/2, but is there a
> > known complete list?

>
> The only possible rational values for sin(p pi)
> when p is rational are 0, +-1, +1/2.
>
> It's slightly easier to think about 2 cos(p pi).
> This equals exp(i p pi) + exp(-i p pi) the sum of
> two roots of unity, and so the sum of two algebraic integers.
> So 2 cos(p pi) would be a rational and an algebraic
> integer, and so also an integer (standard theorem: Z
> is an integrally closed domain). Also 2 cos(p pi)
> is between -2 and 2 inclusive.

Thanks (and to K_h, as well).

That's kind of a boring result, when the only examples you
can think of off the top of your head happen to be the only
examples possible.

Date Subject Author
5/24/12 Daryl McCullough
5/25/12 K_h
5/25/12 Robin Chapman
5/25/12 Daryl McCullough
5/25/12 Brian Q. Hutchings
5/25/12 Brian Q. Hutchings