Maxima doesn't count exact result of sin(%pi/12) or cos(%pi/12), etc. despite the fact it's simple. It is (sqrt(3)-1)/2*sqrt(2) - or (in other form): (sqrt(6)-sqrt(2))/4.
This is really a feature request, not a bug.
Having said that, I have some code to handle this and other cases like %pi/3/2^m and %pi/5/2^m.
There is also a problem in numerical evaluation of sin(%pi/6) and cos(%pi/3):
Why aren't they 0.5? It is not a big error, but it should be fixed
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They are not exactly 0.5 because of the way they're computed. %pi/6 and %pi/3 are converted to double-floats and the Lisp function cl:sin and cl:cos are called to compute the result.
Try float(sin(%pi/6)) and float(cos(%pi/3)). These return exactly 0.5 because maxima simplifies sin(%pi/6) to 1/2, and float converts that to a float.
As written in a posting in this thread, this is a feature request.
Moving this bug report to the tracker "feature requests".