Why is the ratio of two particular sides of the right triangle equal to sine of the angle?

Question

NO COORDINATES, I find them extremely detrimental to conceptual and geometric reasoning.

How did people calculate the angles of triangles from its sides before the trig functions, and why do the trig functions defined as solutions to differential equations and expressed as infinite series have such geometric realization.?

Why exactly does the sine infinite series of the angle give me the ratio of corresponding triangle sides?

How do you avoid coordinates when it is a fundamental fact that the cosine and sine are the coordinates of a point of a given angle on the unit circle? Have you tried to use the search function to find related questions on the relation of the geometric and analytic trigonometric functions, or checked the "Related" list on the right sidebar? Also on mathoverflow and math education/history. — Lutz Lehmann, Mar 01 '20 at 10:49
Your question is too broad, one could fill pages on its aspects. Please add your opinion or observations on: the geometric-numerial approximation of the sine function, the connection of analytical sine fct. and geometric sine ratio is so non-trivial it deserves the name "Fundamental Theorem", an alternative formulation of your question?, question on angle measure and arc length. — Lutz Lehmann, Mar 01 '20 at 12:05
And I would like a simple and precise answer, as simple as it can be but not simoler — Matko, Mar 01 '20 at 12:12
Are you trolling along the skit "It could be trivial or very deep" - "Can't it be both"? One can easily geometrically construct divisions of the unit circle, $15^\circ$ or $3^\circ$, and compute from analytical geometry expressions for their sine. However, to get to arc length, differential equation, Euler formula, power series etc. requires an analytical limit, which is outside geometry and not "simple", https://math.stackexchange.com/a/820655/115115 — Lutz Lehmann, Mar 01 '20 at 12:17
Possible limit arguments: https://math.stackexchange.com/a/2425423/115115 or https://math.stackexchange.com/a/1520674/115115, and again sorry, coordinates can not be avoided for the step "geometry -> analytical geometry -> calculus". — Lutz Lehmann, Mar 01 '20 at 12:31
No. It's not about analogical geometry. It assumes coordinates and things get complicated. — Matko, Mar 01 '20 at 12:41

Why is the ratio of two particular sides of the right triangle equal to sine of the angle?

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