These are my latest notes (June 5, 2009) on the computation of
minimal polynomials of all six trig functions of
angles which are rational multiples of
now have elementary (precalculus) methods for
computing them, but the proofs the process works are
not elementary; relying for example on the
irreducibility of the cyclotomic polynomials which
requires an excursion into quotient fields for
I have scanned
the pages and thumbnailed them here. Click on the thumbnails to
get full images. The manuscript has fourteen
pages, and more will be added if we complete the
association of tangents of classes divisible by 4
with their minimal polynomials.
You may print and disseminate these results without
restriction. If you have questions, please
email me at
and I will try to answer them for you.
I am indebted to Jack Calcut of Michigan State
University for his assistance and references to his
own seminal work related to this problem.