Another amazing work by Hans Boehm back in 1999: a constructive real number calculator, https://www.hboehm.info/crcalc/ . In a sense, this is "as far as a computer can go" in the hierarchy of accuracy that includes floating-point, exact rational numbers, and then constructive reals.
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Another option you see more frequently in algebraic geometry is to use the rational parameterization of the circle, ((1-t^2)/(1+t^2), 2t/(1 + t^2)), which corresponds to the substitution t = tan(theta/2). https://en.wikipedia.org/wiki/Tangent_half-angle_substitution …
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Or you can work directly with x and y subject to x^2 + y^2 = 1 since you're already working with polynomial reduction relations. In either case you get x and y that parameterize a circle and can be used to make a rotation matrix.
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