Famously, it is impossible to trisect an angle with straightedge and compass. One way to prove this—this is how our textbook did it when I took the relevant class—is to show that it impossible to construct a 20 degree angle, by showing, if memory serves me, that the sine or cosine of a 20 degree angle satisfies a certain polynomial whose roots are not constructible. In fact, this impossibility shows a slightly stronger claim than the impossibility of trisecting a general angle. It shows that it is not even possible in general to trisect a constructible angle. For a 60 degree angle is constructible, and if a 20 degree angle is not, then a 60 degree angle is a non-trisectable constructible angle.
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