I am fond of distinguishing between the "pre-rigorous", "rigorous", and "post-rigorous" phases of mathematical education, see
For the "pre-rigorous" stage (which, in the US, is basically everything up to undergraduate calculus), I don't see a pressing need for necessarily introducing and working with a concept (e.g. the sine function) before the rigorous foundations for that concept have been introduced; an informal appeal to Euclidean geometry should suffice at this stage.
Things do get more interesting at the "rigorous" stage (which, in the US, roughly starts at a good undergraduate real analysis class), when students already have plenty of pre-rigorous exposure to real numbers, limits, special functions, etc. but are now ready to revisit these concepts from a rigorous foundational point of view. In my own textbook at this level, I proceed by this route:
- Define rational numbers
- Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences
- Define reals as the space of Cauchy sequences of rationals modulo equivalence
- Define limits (and other basic operations) in the reals
- Cover a lot of foundational material including: complex numbers, power series, differentiation, and the complex exponential
- Eventually (Chapter 15!) define the trigonometric functions via the complex exponential. Then show the equivalence to other definitions.
But certainly one can proceed in a different order to the above.
At the post-rigorous level, one can view of course trig functions as special cases of much more general operations, such as the exponential operation on a Lie algebra...