Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
Trigonometry† is still super important for all of physics and a great deal of higher mathematics.
The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form.
Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry.
The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though.
†: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.
