Updates:
Labels: -Type-Defect Type-Enhancement Polynomial
Comment #3 on issue 2949 by [email protected]: trig functions do not
return exact results for half angles
http://code.google.com/p/sympy/issues/detail?id=2949
Well, that's not a big deal, because you can compute it from the formulas,
and it's more useful (at least in many cases) to have it in terms of an
algebraic expression. For example, right now, minpoly() will only work if
it's in an algebraic form (see issue 2789).
Apparently, sine or cosine of any rational multiple of pi is algebraic, so
can be represented either by a radical expression or using RootOf. This
also applies to exp(I*r*pi), with r rational, by Euler's formula. So we
should implement expands for all of these.
But perhaps we should put a limit on auto-expanding only simple fraction
multiples of pi (say, only integer multiples of pi/12 or pi/24), and only
expand others with expand_trig().
Question: is our current root-finding code sufficient so that we can just
compute the minimal polynomial (using the facts about roots of unity), and
then compute the formal root, or do we need to compute it recursively with
the half angle formulas? If it's the latter case, then we should use this
to improve our root finding algorithm.
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