geometry - To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things and starts talking about angles all of the sudden, or uses a contrived definition like the unique $\theta\in[0,\pi]$ such that $\|u\|\|v\|\cos\theta = u\cdot v$. Yes it works fine, but it leaves me quite unsatisfied; I should like to already have an "intrinsic" definition of an angle, and then let the cosine function be defined to tell me things about angles. Why, the other way around we have to define cosine by some magical power series, and then it turns out that the above definition makes angles behave as they should! To me it seems disingenuous, but thoughts on this are welcome.

Anyway, the construction comes from a book called Àlgebra Lineal i Geometria (Castellet/Llerena), and I'd like to know if anyone has seen it, other than in this book of course. I'll post the beginning of the section (translated from Catalan, and paraphrased):

Let $(E,\langle\cdot,\cdot\rangle)$ be a $2$-dimensional Euclidean space. In the set of pairs of unit vectors, define the equivalence relationship $$(u,u')\sim(v,v') \iff \exists f\in SO(2) : f(u)=v,\, f(u') = v'$$
This condition is proven to be equivalent to $\exists g\in SO(2) : g(u)=u',\, g(v) = v'$. We define an angle as one of these equivalence classes. We denote the class represented by $(u,u')$ as $[(u,u')] = \widehat{uu'}$. This can be easily extended to $E\times E$ by defining $\widehat{uv}$ as the angle defined by $\frac{u}{\|u\|},\frac{v}{\|v\|}$. Call the set of angles $A =E\times E_{\large{/\sim}}$ and for any $u\in E$ define a map $$SO(2)\longrightarrow A \atop \qquad\qquad\, f\mapsto \widehat{uf(u)}$$ This is in fact a bijection, and allows us to transport the operation in $SO(2)$ to $A$: given $\alpha,\beta\in A$ with preimages $f,g$ respectively, define the sum $\alpha+\beta$ as the image of $f\circ g$. In class notation: $$\left.\begin{align}&\alpha = [(u,f(u))] \\ &\beta = [(f(u),gf(u))]\end{align}\right\}\Rightarrow \alpha+\beta = [(u,gf(u))]$$
Naturally, this sum has the same properties in $A$ as does the operation in $SO(2)$. $A$ is then an abelian group, whose identity is $0 = \widehat{uu}$. The inverse, or opposite angle of $\widehat{uf(u)}$ is $\widehat{f(u)u}$.

Here comes the fun part.

By fixing an orientation on $E$, each $f\in SO(2)$ has a corresponding matrix $$\left(\begin{array}{ccc}a & -b \\ b & a\end{array}\right)\text{ with } a^2+b^2 = 1$$ Let $\alpha$ be the angle corresponding to $f$. We define the cosine, and the sine, of $\alpha$ by $$\cos\alpha = a\qquad \sin\alpha = b$$

Too clever. And it gets better:

We'll make a couple of observations now about this definition. Firstly, in changing the orientation of $E$ the sign of $\sin\alpha$ changes, but not $\cos\alpha$.

One has $$\cos0 = 1\qquad \sin0 = 0$$ since the angle $0$ corresponds to $\mathrm{id}$.

The angle $-\alpha$ (the opposite wrt the sum) corresponds to $f^{-1}$, whose matrix is the transpose of $f$'s matrix; therefore: $$\cos(-\alpha) = \cos\alpha\qquad \sin(-\alpha) = -\sin(\alpha)$$

The angle $\alpha + \beta$ corresponds to the composition of their respective maps. Thus, matrix multiplication gives: $$\begin{align}\cos(\alpha+\beta) = \cos\alpha\cos\beta-\sin\alpha\sin\beta \\ \sin(\alpha+\beta) = \sin\alpha\cos\beta+\cos\alpha\sin\beta\end{align}$$

To finish I'll put some subsequent propositions without proofs.

There exists one, and only one angle $\pi$ such that $\pi+\pi = 0$. $\pi$ is the angle such that $\cos\pi = -1$ and $\sin\pi = 0$.

There exist two, and only two angles $\delta_1,\delta_2$ such that $\delta_i + \delta_i = \pi$. $\delta_i$ are the angles such that $\cos\delta_1 = \cos\delta_2 = 0$ and $\sin\delta_1 = -\sin\delta_2 = 1$. We call these right angles.

$\widehat{uv}$ is a right angle iff $\langle u,v\rangle = 0$

The text continues proving things like these. My question is whether anyone has seen this, or a similar extensive treatment. Also though, I'm interested in finding any text that formally links the primitive angles, sine and cosine from geometry to the sine and cosine we now all know and love from calculus, or even complex analysis, preferably with geometry as a starting point.

Notes:

• Obviously there are well definedness issues to address. It seems this is left to the reader.


• Should I post this on mathoverflow? I've never used it but something tells me a bibliographic inquiry like this one could fit.



• Unfortunately, angles are now abstract objects, and we haven't at all defined the sine or cosine of a real number, so I'm thinking of a way to map real numbers to angles. Any comment on this would be appreciated!