It's not really clear what "learning trig" actually means.

Trigonometry, as a subject, can be fully understood in a sentence: "Triangles (especially triangles with a 90-degree angle) have special properties that make them interesting." This is not useful, of course, and worse, it fails to justify high schoolers spending most of a year on the subject.

Trig might instead be understood in this way: "Breaking down complex shapes into triangles makes it possible to find many of the values involved in the shape, because of the special properties that triangles have."

I think this is a fair summary of the topic. If that's good enough, it might be worth doing an article based around taking a complex collection of lines and curves (and draw it out of a photograph for making it feel applicable), assigning a length here and there, and then spending the entire article zooming in on specific sections in order to figure out the consequent lengths of everything else.

I don't think you can get away describing trig in a sentence without using the word 'circle'. Really I'd argue trig is at its core more about the circle.

But it's trigonometry, not anometry.

Really, the unit circle is important because it's the unit circle, not because it's the unit circle. A circle is 2π angles. A triangle is composed of the same kind of angles, which means that, because the diameter of the circle doesn't matter (because it's a unit circle), then it's less important what the lengths are than it is how the lengths relate to one another. I.e., what factor of a unit vector as compared to what other factor of a unit vector.

What's a circle if not all the right triangles (up to a choice of units)?

The solution set to equations of the form X^2 + Y^2 = R^2?

The boundary of a ball in a 2-dimensional, euclidean metric space?

An equivalence class of Gaussian numbers based on their norms?

Look at the beauty of the sin function and tell me that has more to do with triangles than the circle. sin and cos are the length of the projected vector along each axis as a point moves about the circle. connecting those points forms a triangle, sure, but it seems less fundamental.

I'd absolutely concede that the sine function is more about a circle than it is about a triangle. If you could really say that the sine function is about a geometry at all.

Thanks for the feedback! I'm in general agreement with your second summary; basically, the properties of triangles end up appearing all over math, so our trig terminology/relationships become generally applicable, especially to circles and other repeating patterns.

One of the best parts of trig is that knowing one little fact (sin(x) = foo) reveals a tremendous number of other ones (inverse sine, get the original angle; then the cosine, tangent, secant, etc.). I'd like to explore some applications (geometric and non), appreciate the suggestion!

I like pointing out to students of classical mechanics that the first year is all about pointing out ramifications of one assumption, one mindlessly simple, underwhelming equation: constant acceleration. If you know a bit of calculus. It goes from being a terrifyingly large amount of memorization and calculation to... "Oh."