Given any angle how can you say that it is constructable or not?
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According to wiki link given by J.M an angle is constructible if its sine, cosine or tangent is constructible. So given an angle if cosine of the angle satisfies an irreducible polynomial of degree which is NOT a power of 2 then it is NOT constructible. The proof of this criteria is standard (For example you may find it in Hersteins Topics in Algebra, section 5.4)