so according to what you just said, trigonometric functions were useless before discovering the half, double, and addition identities? that's what got me baffled in the first place, how would they be useful without knowing angles other than 0-30-45-60-90 and maybe 36 and 72. You said it might be by measurement but would they really rely on measurement for a good amount of time before finding the identities?
No, I don't think I said anything about measuring anything.
Essentially, what I said is that, whatever development the concepts went through initially, the necessary identities were known by the time anyone did anything elaborate with them, because the latter required the tables. We might say (sort of reversing what you just said) that the discovery of various identities was what made the trig functions
useful, so either the identities were discovered rather quickly (after all, they knew lots of geometry), or it was the identities that led to using trig functions. I'm not sure that any trig functions were even thought of as things worth naming until identities were known and perhaps until tables were created. (Keep in mind that the very concept of "function" is a relatively recent idea (1800's); a table may have been the closest concept they had. And they had none of the algebraic symbolism we have now.)
Perhaps you have an inaccurate picture of the process of developing a field of mathematics. The sine wasn't lying around somewhere with people wondering, "How can we use this thing?" Rather, people gradually developed theorems about ratios of sides of triangles, discovered identities that could be used to calculate them, thought of problems that could be solved if only they had tables of values, and then made the tables. Only then was there a subject of trigonometry that could be studied. (And the sine, itself, was not given a name until the 500's AD.)
But that is still a vast oversimplification. If you've read the pages I referred you to, you might want to move on to broader stories of the history of trigonometry. Read the whole Wikipedia article, and maybe others such as
this and
this. It's also worth considering that some of the geometry that Euclid wrote about is equivalent to bits of trigonometry, but without identifying the functions. For example,
this theorem is really the Law of Cosines.