The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is:
$$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-295364007592722432 x^{38}+ \\ +411985976135516160 x^{36}-452180272956309504 x^{34}+396366279591591936 x^{32}- \\ -280058255978266624 x^{30}+160303703377575936 x^{28}-74448984852135936 x^{26}+ \\ +28011510450094080 x^{24}-8500299631165440 x^{22}+2064791072931840 x^{20}- \\ -397107008634880 x^{18}+59570604933120 x^{16}-6832518856704 x^{14}+ \\ +583456329728 x^{12}-35782471680 x^{10}+1497954816 x^8- \\ -39625728 x^6+579456 x^4-3456 x^2+1$$
It obviously has $48$ roots, but since it's even we only need to consider $24$ positive roots.
One is $x=\sin 1 ^\circ=\cos(\frac{\pi}{2}-\frac{\pi}{180})=\cos \frac{89\pi}{180}=\cos 89 ^\circ$.
It seems that all the other roots are made using numbers of degrees $<90$, which share no common divisors with $180$:
$$x=\cos \alpha ^\circ$$
$$\alpha=\{1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$
What is the general algebrac reason for this? How does this rule work for other trigonometric functions of rational multiplies of $\pi$?
If, in general, we find a polynomial for the following number:
$$y=\text{trig} \frac{p}{q} \pi $$
Where $\text{trig}=\{ \sin, \cos, \tan \}$, $p,q$ - integers, then what will the other solutions be?
In A Note on Trigonometric Algebraic Numbers by D. H. Lehmer (and also in Wikipedia), we find that $$ z^{-d}\Phi_n(z) = \Psi_n(z+z^{-1}) $$ where $\Phi_n$ is the $n$-th cyclotomic polynomial and $d=\frac{\phi(n)}{2}$ is the degree of $\Psi_n$, which is half the degree of $\Phi_n$. Lehmer proves that $\Psi_n$ is irreducible.
The roots of $\Psi_n$ thus correspond to pairs of roots of $\Phi_n$, which are the primitive $n$-roots of unity. Therefore, the roots of $\Psi_n$ are $2\cos\left(2\frac{k\pi}{n}\right)$ with $\gcd(k,n)=1$.
Thus, the numbers $\cos\left(2\frac{k\pi}{n}\right)$ are the roots of $\Psi_n(2x)$.