I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it?

When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant.

I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...