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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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 quasi Posts: 12,042 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 12, 2013 7:10 AM

Richard Tobin wrote [edited]:
>
>If you could find an angle 0 < a < 30 such that sin(a) and
>sin(30-a) are both rational, you could add some sides to
>a regular hexagon.

By the law of cosines, cos(a) and cos(30-a) would also have
to be rational.

But then, letting b = 30-a,

cos(30) = cos(a)cos(b) - sin(a)sin(b)

contradiction, since the LHS is irrational while the RHS is
rational.

Thus, no such angle a exists.

quasi

Date Subject Author
12/10/13 quasi
12/10/13 ross.finlayson@gmail.com
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12/11/13 quasi
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12/12/13 quasi
12/12/13 Helmut Richter
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12/11/13 scattered
12/11/13 quasi
12/11/13 fom
12/11/13 fom
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12/11/13 Richard Tobin
12/11/13 Richard Tobin
12/12/13 quasi
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12/12/13 Thomas Nordhaus
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12/13/13 Richard Tobin
12/13/13 Richard Tobin
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12/13/13 Richard Tobin
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12/15/13 quasi
12/15/13 Richard Tobin
12/15/13 David Bernier
12/15/13 quasi
12/18/13 Richard Tobin
12/18/13 ross.finlayson@gmail.com
12/19/13 quasi
12/14/13 Richard Tobin
12/14/13 quasi
12/14/13 Richard Tobin
12/14/13 ross.finlayson@gmail.com
12/15/13 Brian Q. Hutchings