This is about to be closed as "unclear what you're asking", perhaps I misunderstood the question, but it seems you ask for a characterization of the rational angles $\alpha=2\pi k/l$ that produce an irrational cosine. This question is addressed by Jörg Jahnel in these notes, see in particular section 6: $\cos\alpha$ is an algebraic number of degree $d>1$ if and only if the Euler function $\phi(l)=2d$.

This is about to be closed as "unclear what you're asking", perhaps I misunderstood the question, but it seems you ask for a characterization of the rational angles $\alpha=2\pi k/l$ that produce an irrational cosine. 

This question is addressed by Jörg Jahnel in these notes, see in particular section 6: $\cos\alpha$ is an algebraic number of degree $d>1$ if and only if the Euler function $\phi(l)=2d$. For example, a quintic irrationality is obtained only for angles $\alpha/2\pi=1/22$ or $5/22$.

this is about to be closed as "unclear what you're asking", perhaps I misunderstood the question, but it seems you ask for a characterization of the rational angles that produce an irrational cosine.

This is about to be closed as "unclear what you're asking", perhaps I misunderstood the question, but it seems you ask for a characterization of the rational angles $\alpha=2\pi k/l$ that produce an irrational cosine. This question is addressed by Jörg Jahnel in these notes, see in particular section 6: $\cos\alpha$ is an algebraic number of degree $d>1$ if and only if the Euler function $\phi(l)=2d$.