Generalizing Odom's construction of the golden ratio

Question

The artist and amateur mathematician George Odom found this nice construction for the golden ratio $\phi$ using an equilateral triangle and its circumcircle,

$\hskip2.3in$ $\hskip3.3in$Fig. 1

Let $A$ and $B$ be the midpoints. The ratio of the line segments $|AB|$ and $|BC|$ is, $$R_3 = \frac{|AB|}{|BC|} =\phi$$

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We can easily generalize the above figure using a square,

$\hskip2.3in$ $\hskip3.3in$Fig. 2

Let the (sadly invisible) $A,B,C$ of Fig.2 be analogous to that of Fig.1. What is then its,

$$R_4=\frac{|AB|}{|BC|} =\,?$$

Q: In general, is $R_n$ for $n>3$ algebraic? If it is, does its minimal polynomial have a closed-form?

Answer 1

From purely trigonometric considerations (see this Desmos simulation, or ask if you need clarifications), I've found a closed form for your sequence: $$R_n = \frac{\sin\left(\frac {2\pi}{n}\right)}{\sqrt{1-\left(\frac{1+\cos\left(\frac{2\pi}{n}\right)}{2}\right)^2}-\frac 1 2\sin\left(\frac {2\pi}{n}\right)}$$

It does not look pretty, but I'm sure it can be simplified: I might get back to this when I have more time in my hands. Also have a look at the answers to this question if you want to prove that $R_n$ is algebraic for every $n$.