No. There's the analytic definition of sin that does not mention curves or lengths at all: the sine is the unique function that satisfies the following differen

No. There's the analytic definition of sin that does not mention curves or lengths at all: the sine is the unique function that satisfies the following differential equation and initial value problem: s''(x) + s(x) = 0 s(0) = 0 s'(0) = 1 It's a nifty way to define sine purely by its differential properties. Of course, it requires some work to show that differential equations have a solution and that this particular solution is sine (i.e. has the properties you want a sine to have), but once you do that work, it's pretty easy to prove things such as sin^2(x) + cos^2(x) = 1. On the other hand, starting from the geometric definitions (and building the framework for that, such as arclength, which really requires calculus), it takes a longer route to get to the calculus of sine. Historically this was the route, but we have found shortcuts since then.