Will science end?

[Radar]

Geometry certainly isn't finished, because (as Euclid showed) you can use geometry to do number theory, and there are plenty of open questions in number theory.

For trigonometry specifically, last year a couple of high school students published a new proof of the Pythagorean Theorem using the Law of Sines, just because they had read in a book that it was impossible to do so.

You can also do everything a quantum computer does using geometry:
https://arxiv.org/abs/quant-ph/0611279v2

There is a whole field of geometric algebra that people actively publish in.