quasi
Posts:
11,708
Registered:
7/15/05


Re: rational ngon 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(30a) are both rational, you could add some sides to >a regular hexagon.
By the law of cosines, cos(a) and cos(30a) would also have to be rational.
But then, letting b = 30a,
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

