i want to argue that the goal is approximating pi without making use of “trig functions” is in some sense hopeless. Indeed these functions are more naturally called circular functions, and to me this is like asking for the length of a semicircle without using properties of the circle. Indeed the sine function evaluated at t is just the y coordinate of the end point of the counterclockwise arc of length t starting at (1,0), Hence arcsin(y) is by definition the length of the counterclockwise arc from (1,0) to the point of the unit circle with second coordinate y. Since thus arcsin is precisely the arclength function, asking for the value of π is the same thing as asking for the functional value 2arcsin(1), (or arccos(-1)).
The OP made a valid point in observing that it seems problematic to try to evaluate trig functions at rational multiples of π when one does not know in advance how to approximate π. But this is beside the point here, since it is the inverse trig functions which are used, and it is precisely because one often obtains rational outputs from trig functions whose inputs are rational multiples of π, that conversely, one obtains rational multiples of π as outputs of inverse trig functions whose inputs are rational numbers. I.e. using inverse trig functions, one does not need to know approximations to π, rather one plugs ordinary rational numbers into power series, such as that named after Leibniz, (but discovered earlier it seems by Madhava), and the partial sums of the outputs give approximations to π.
On the other hand, if one uses the method of Archimedes, approximating the length of the semi circle by the lengths of tangent segments, one is indeed using ordinary trig functions, not their inverses, evaluated at rational multiples of π, but these inputs are done by geometry, i.e. by subdividing a semicircle into an integer number of parts, and then evaluating by actual measurement. This method uses the fact that the function tan(πx)/x converges to π as x—>0.
I.e. rather than trying to plug πx into a power series for tan, we subdivide a unit radius semicircle into n equal parts, draw a right triangle with vertex at the center of the unit circle, right angle vertex at (1,0), and acute angle = π/n at the center (0,0), hence with hypotenuse dividing the semi circle by n, then measure the vertical side opposite the angle π/n, and approximate π by n times this length. I.e. this side opposite the angle of π/n has length tan(π/n), and the approximating polygon to the upper semicircle has length n.tan(π/n), which approaches π as n approaches infinity.
So I claim all approaches to approximating π must involve trig functions one way or another, even series like π^2/6 = SUM (1/n^2), as Euler explains in his wonderful little book “On the analysis of infinities”, chapter X, at least if you notice that (e^t - e^-t)/2 = sinh(t) = i sin(t/i), where i = sqrt (-1). Euler also states in chapter VIII that he himself approximated π to hundreds of digits, using the series for arctan evaluated at 1/sqrt(3), "with incredible labor", and then explains how to greatly reduce this labor by using the addition formula for tan.here is a nice explanation of how Archimedes method can be done in practice, without physical measurement, by using "double angle" trig formulas:
https://arxiv.org/pdf/2008.07995.pdf