One learns trigonometry in high school/secondary school and either forgets it if one continues onto a career less mathematical or, possibly, uses it extensively in their work, as do engineers and physicists.
As a field of study in mathematics however, it seems that trigonometry is mostly "solved", at least it seems so for the familiar trigonometry in $\mathbb{R}^2$. Is this true, or are there still interesting questions that deal with trigonometry or, perhaps, generalizations of it?
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Trigonometry on its own is for the most part no longer a very active area of research; though trigonometry remains a key tools in applied research today (for instance studying the rigidity of origami patterns). And the trig functions as eigenfunctions of the Laplacian have countless generalizations and applications.
One case I've seen of modern fundamental research on trigonometry is Wildberger's work on rational trigonometry, which seeks to reformulate trigonometry as purely algebraic relations of positions (see his web site). But NB that while as far as I can tell this work is sound, it is at the fringes of mainstream mathematical research.