Having had a non-traditional route through learning various math topics, I'd like to advocate the teaching of calculus (the understanding of the accumulation of infinitesimally-small changes) before the teaching of trigonometry (the relationships between different geometric shapes, and the resulting association with waves).
In my (humbly biased) opinion, it's easier to learn how trigonometric concepts operate with a calculus background, than with merely an algebra-and-geometry background. Trigonometry complicates calculus education - knowing trigonometry may facilitate better calculus understanding to some, but not knowing trigonometry doesn't necessarily complicate learning calculus concepts. The trigonometry-first tradition in math education reflects a "number-phile" bias, not the most optimal route for humans to ingest and retain concepts.
Edit: The "number-phile" bias is clear - we want our teachers to be people with passion on the topic, as we're more likely to get a better understanding of the topic. Understanding sine waves as the result of an integral was much easier for me to understand than as the result of a complicated calculation.
It’s somewhat misleading/unfortunate to use angle measures as the basic characterization of an angle/rotation, when just trying to solve basic geometry / triangle measurement problems. It’s almost always easier to use a vector representation, written in terms of explicit coordinates if necessary (e.g. as a complex number). http://www.shapeoperator.com/2016/12/12/sunset-geometry/
Where angle measures and sines/cosines start to be useful is with uniform circular motion, which as you say is a calculus problem.
The key to understanding the form of trigonometry courses is to understand the context for their creation. Namely, all calculations (e.g. for astronomy, navigation, engineering, mapmaking, ...) used to be done by hand by humans, which was very expensive. It was important to have someone with fluent knowledge of trigonometric identities simplify formulas to a form with as few arithmetic operations and table lookups as possible to save money, or just to match the available function tables. Today in a computer age, extensive memorization of trigonometric identities is an anachronism, and spending lots of time on practicing their manipulation is okay algebra practice but not anything directly useful per se.
In my (humbly biased) opinion, it's easier to learn how trigonometric concepts operate with a calculus background, than with merely an algebra-and-geometry background. Trigonometry complicates calculus education - knowing trigonometry may facilitate better calculus understanding to some, but not knowing trigonometry doesn't necessarily complicate learning calculus concepts. The trigonometry-first tradition in math education reflects a "number-phile" bias, not the most optimal route for humans to ingest and retain concepts.
Edit: The "number-phile" bias is clear - we want our teachers to be people with passion on the topic, as we're more likely to get a better understanding of the topic. Understanding sine waves as the result of an integral was much easier for me to understand than as the result of a complicated calculation.