Certainly, you can check equivalence with {pi/3+2*pi*n, -pi/3+2*pi*n} and if that test fails you can then check for equivalence with {pi/3+2*pi*n, 5*pi/3+2*pi*n}. That might work in this case.
I think what you are really trying to test is equivalence with the union
\[ \{\pi/3+2n\pi| n \in \mathbb{Z}\} \cup \{-\pi/3+2n\pi| n \in \mathbb{Z}\}. \]
I'm not aware of any computer algebra systems which can robustly do this in general!
Chris,
I may have (in my mind at this moment) a solution to check (in Maxima) the equivalence of two sets (as the general solution set of a trigonometric equation), each of them of this kind:
{a1+b1*n, a2+b2*n, ...,ak+bk*n}
where b2/b1, b3/b1, ..., bk/b1 are all rational numbers, n is an integral variable and k is a finite natural number. All numbers (a1,b1,a2,b2, etc. and k) may be different for these two sets. Only n is the common variable.
Please let me know if you are interested.
I may send you either the algorithm or the Maxima code.
Teo
