[replicant]

As young students, we are taught that these functions are ratios of sides on right angles. It is fairly intuitive for them that provided they can draw a right triangle, they just need to measure the sides. And it is easy to accept that someone computed sines and cosines for a lot of angles and put it in a table. My point being, that we didn't need (in the past) calculus to compute trigonometric functions, and I don't see how it is a burden for students to be introduced to those functions without the Calculus definition.

And the reality is, that the definition of sine as a ratio of the catheti and hypotenuse is a rigorous definition of the function. Strictly, this sine is different from the sine of calculus. The first, the sine from Euclidean geometry, assigns a real to pair of rays, while the calculus sine, is function from the real numbers to the reals. And it does take some work to link them formally.