Definition talk:Angle

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Angles as equivalence classes

Is there a reference worthy approach somewhere that treats angles as equivalence classes? The equivalence relation being on $(\mathbb{R}^2)^3$. Likewise, for solid angles an equivalence relation on $(\mathbb{R}^3)^4$. --Jshflynn (talk) 17:50, 31 October 2012 (UTC)

I think I comprehend what you try to say. So $(x,y,z) \sim (x',y',z')$ iff the angles $yxz$ and $y'x'z'$ are equal. I haven't needed solid angles for a long time now, so I'm not entirely sure how you interpret $(\R^3)^4$. --Lord_Farin (talk) 19:54, 31 October 2012 (UTC)
Yes that's right. In the construction of my analysis notes I've been trying to avoid the inevitable. I dislike the analytic definition of sine and cosine because it conceals their motivation. Tracing it back, the motivation ultimately depends on the sin(x)/x theorem. For this we have the geometric proof. But that's where it becomes difficult (for me at least) to avoid circularity and synthetic geometry. Deciding on what the most fundamental thing is and then proceeding from there. I've been playing with this idea, the hypervolume axioms and adding in an axiom to do with isometries which is better when angles are defined like this. I don't want to cave in but I think I ultimately must to keep with my schedule. The analytic definition it is then... --Jshflynn (talk) 19:00, 1 November 2012 (UTC)
"I dislike the analytic definition of sine and cosine ..." I rather fear that you may be out on a limb. Ultimately the analytic route is the main way in from the axiomatic framework starting at Zermelo-Fraenkel. A full axiomatization of mathematics starting from the geometric foundation of e.g. Tarski or Euclid is still embryonic on this site (only because I lose patience with fiddly diagrams), and (not having studied it fully) I'm not sure can be done without at least passing through the definition of the trig functions as infinite series.