geometry - Have "algebraic angles" been studied before?

I'm writing a geometric software library and I came up with a useful concept.

Let's call a real number $\alpha$ an algebraic angle if $\alpha\in[0,2\pi)$ and $\cos \alpha$ is an algebraic number. The set of algebraic angles has some pretty neat properties:

• The sine, cosine, and tangent of an algebraic angle are algebraic.


• Define negation and addition of angles the usual way, wrapping around at $2\pi$. Then algebraic angles are closed under negation and addition (since $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$).


• Define multiplication of an algebraic angle by a rational number to also wrap around at $2\pi$. Multiplying an algebraic angle by a rational yields another algebraic angle! This can be proved from the identity $\cos(n\alpha)=T_n(\cos\alpha)$, which I learned about here. $T_n$ is a Chebyshev polynomial of the first kind. In particular, it is a polynomial with integer coefficients.


• Since $\pi$ is an algebraic angle, so is any rational multiple of $\pi$ in $[0,2\pi)$.



• Algebraic angles are a vector space over the rationals (I haven't proved anything interesting using this fact though). They're not a vector space because $x(y\alpha)$ is not necessarily equal to $(xy)\alpha$. For example, $(1/2)(2\cdot\pi)$ is $0$ but $(1/2\cdot 2)\pi$ is $\pi$.

Has the set of algebraic angles appeared in academic literature? Individual algebraic angles come up everywhere in geometry: for example, the interior angle between two faces of a regular dodecahedron is $\arccos(-\frac{1}{5}\sqrt{5})$.

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Let me elaborate on why an algebraic angle divided by an integer yields an algebraic angle: Suppose that $\cos\alpha$ is algebraic, so that there is some polynomial $P(x)$ with integer coefficients such that $P(\cos\alpha)=0$. Let $n$ be a positive integer. By the Chebyshev identity above, $P(T_n(\cos(\alpha/n)))=0$, so $\cos(\alpha/n)$ is algebraic.

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Edited 2017-01-10:

Ok, after reading some of the answers and comments I realized that the interesting properties of algebraic angles become obvious if you consider how they transform under $\alpha\mapsto e^{i\alpha}$. The algebraic angles become algebraic points on the unit circle, angle negation becomes complex conjugation, angle addition becomes complex multiplication, and multiplying by a rational essentially becomes raising to a rational power. It's clear that algebraicity is preserved by each of these operations.