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What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and series realizations.

This is usually done by calculating arclengt in a coordinate system but I want to avoid completely any explicit or implicit use of coordinates.

Since a circle is a set of all points equidistant from a given point, it follows that it is a limit of a sequence of regular n sided polygons as n goes to infinity

How do I show that a half lenght side of one n sided poligon is the sine function of pi over n? Without in any way introducing coordinates

What way would there even be to define or calculate pi in such an axiomatic coordinate less setting

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    $\begingroup$ see en.wikipedia.org/wiki/Synthetic_geometry ; isn't that what you are looking for? $\endgroup$ – Carlo Beenakker Apr 9 '20 at 8:22
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    $\begingroup$ So i have voted to reopen. $\endgroup$ – Konstantinos Kanakoglou Apr 11 '20 at 21:10
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    $\begingroup$ @KonstantinosKanakoglou I did not vote to close, but I found the wording of the question unclear, and I couldn't get a good idea of what would satisfy the OP as being "co-ordinate free" -- see mathoverflow.net/questions/352362/… and math.stackexchange.com/questions/3484627/… for context $\endgroup$ – Yemon Choi Apr 12 '20 at 4:55
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    $\begingroup$ It doest matter what I consider coordinate free(thaugh I explained it clearly in all my posts those lines if for some reason that is of interest here.. I staded it perfectly unambiguously here too look at the description:"without in any way introducing coordinates",... $\endgroup$ – Ezio Apr 12 '20 at 9:08
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    $\begingroup$ There are nice characterizations/definitions of the trig functions in terms of functional relations; these do not make any direct reference to any kind of coordinates. I believe the most well known is the following one: Let $p$ be a real number and $C(x)$, $S(x)$ be real valued functions of a real variable satisfying the following conditions: (a). $C(x-y)=C(x)C(y)+S(x)S(y)$ for all real $x,y$, (b). $S(p)=1$ and (c). $S(x)\geq 0$ for all $x\in [0,p]$. Then $S$ and $C$ are uniquely determined. $\endgroup$ – Konstantinos Kanakoglou Apr 14 '20 at 19:11
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Trigonometric functions do not belong to geometry. Neither does the "measurement of angles" by real numbers. They belong to analysis. This fact is discussed in detail in the book of Dieudonne, Linear algebra and geometry, Houghton Mifflin Co., Boston, Mass. 1969. (French original: Algèbre linéaire et géométrie élémentaire. (French) Enseignement des Sciences, VIII Hermann, Paris 1964.)

Shortly, the situation is the following: in geometry the group $SO(2)$ is defined. Then, when the ground field is $R$, there is a continuous homomorphism $R\to SO(2)$, which is called the exponential. Then trigonometric functions are defined as matrix elements of this homomorphism. Existence of this homomorpsism is proved using Calculus. When normalized so that derivative at $0$ equals $1$, the positive generator of the kernel is called $2\pi$. Then one obtains power series etc. An especially elegant exposition is obtained when one uses complex numbers. See Whittaker Watson, vol. I, Appendix, or Ahlfors, Complex Analysis, or Bourbaki, Topology.

Remark Interestingly, the ancient Greeks apparently understood this somehow. For them trigonometry and angle measurement were not parts of pure Geometry. Greek mathematicians never used them in their pure geometric investigations. For Euclid and his followers, the existence of an angle of $2\pi/7$ radian is dubious, since he cannot construct it. Trigonometry and angle measurement were parts of applied mathematics (astronomy, geography, geodesy). In medieval times this distinction was blurred, and it is still blurred in education.

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    $\begingroup$ Norman Wildberger has shown that trigonometry can be developed without using analysis, albeit by avoiding the notion of an angle $\endgroup$ – ogogmad Apr 10 '20 at 8:45
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    $\begingroup$ I agree that distinction is blurry, but does that mean that in the theory of pure geometry the ratio of circle length and radius is an undecidable question? $\endgroup$ – Ezio Apr 10 '20 at 15:31
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    $\begingroup$ @Alexandre Eremenko Also there are ways to have a rotation group representation without matrices or hyper complex numbers. Every rotation is a composition of two reflections, and in some systems, variations of Clifford algebra where this ratation groups are equivalent to spinors in a completely coordinate free way. $\endgroup$ – Ezio Apr 10 '20 at 15:36
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    $\begingroup$ @ogogmad What Wildberger has and hasn't "shown" seems rather debatable, once one strips away all the ultrafinitist polemic $\endgroup$ – Yemon Choi Apr 11 '20 at 13:42
  • $\begingroup$ @Ezio How do you propose to define the "length" of a circle? There is a reason why Archimedes used the principle of exhaustion $\endgroup$ – Yemon Choi Apr 11 '20 at 13:43

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