Thanks for this. I guess this goes to show how hard it can be to communicate math well. When I learned the sin(x)/x limit I accepted the "proof" by geometric intuition with no protest and was not alert to any deeper source of confusion here.

Come to think of it, the rigorous treatments of sine that I've seen probably all use power series definitions. To see that it's the same function as the one defined using triangles I expect you have to appeal to derivative properties, so that approach would not skirt the issue.