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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