Amhuxt.446
net.math
utzoo!decvax!harpo!zeppo!wheps!eagle!mhuxt!rwhaas
Thu May 6 12:30:24 1982
Re:
If theta is a rational multiple of pi, then sin(theta) and cos(theta)
are algebraic.
First note that
cos(p*pi) = (-1)^p = a q-th degree polynomial in cos(p*pi/q) with
integer coefficients, so that cos(p*pi/q) is algebraic for all
rational numbers p/q.
Since cos(p*pi/q) = 1-sin^2(p*pi/2q) we have the representation
(-1)^p = a 2q-th degree polynomial in sin(p*pi/2q) with integer
coefficients, so that sin(p*pi/2q) is algebraic for all rational
numbers p/2q, or in other words for all p/q since p and q are
arbitrary integers.
Roy Haas
Bell Labs Indian Hill
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