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.
The boundary of a ball in a 2-dimensional, euclidean metric space?
An equivalence class of Gaussian numbers based on their norms?
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!
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.