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Rational values of sine

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Daryl McCullough

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May 24, 2012, 9:32:41 PM5/24/12
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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?

K_h

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May 25, 2012, 1:47:33 AM5/25/12
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"Daryl McCullough" wrote in message
news:30126780-4803-4801...@googlegroups.com...
I hope this helps.

http://en.wikipedia.org/wiki/Niven%27s_theorem

+

Robin Chapman

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May 25, 2012, 4:31:45 AM5/25/12
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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.

Daryl McCullough

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May 25, 2012, 5:48:24 PM5/25/12
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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.

1treePetrifiedForestLane

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May 25, 2012, 5:54:02 PM5/25/12
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that is nice.

1treePetrifiedForestLane

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May 25, 2012, 6:16:17 PM5/25/12
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99999.999994999999999874999999994

allgood; I can clip from the calculator;
that integer was not a perfect second power,
nor "skware."
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