[kragen]

What do those tables say for 59°59'? I'm skeptical that what you're looking at is, strictly speaking, a trigonometric table.

If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexagesimal) fraction approximation, and I guess you're right that it isn't clear that one is necessarily better than the other.

Incidentally, the fact that we write things like 59°59'30" comes about because the Babylonians at least weren't using Wildberger's "spreads" all the time.

[dhosek]

Those tables don’t give a value for that. They only give values for angles whose trigonometric values can be expressed in terms of rational expressions with radicals (and angles as rational expressions in terms of π).

[kragen]

That sounds like a sort of "trigonometric table" you couldn't use in practice for calculation. You need to be able to look up whatever value you measure with your sextant or theodolite to within its measurement precision. You seem to be using the term "trigonometric table" in a way conflicting with the standard use explained in https://en.wikipedia.org/wiki/Trigonometric_tables perhaps as a sort of pun or joke, or perhaps as a sort of act of activism against hegemonic notions of trigonometry.