[replicant]

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)...

[threatofrain]

I think I wasn't clear in saying that I believe the current pedagogical value of trigonometry is in giving students a brief familiarity with the trig functions when they see it again in the context of physics or engineering. Or standardized testing. I think those are the likely scenarios where students are going to be seeing relevance in trigonometry.

What other foundation or learning pathway do you see trigonometry serving as? Somebody else mentioned that it gives students a sense of applications, so they know that calculus is not for nothing. So then I question: what applications? And I pose, how about statistics?

[east2west]

It may be because of your math education but I was introduced to trigonometry in junior high in China. I had, and my classmates had, no trouble understanding them as functions of angles coming from ratios. It is the glue that binds circles and triangles and squares and ... . By high school analytic geometry greatly expanded their scope and use. This is the problem I saw in American high school when I moved to US: shallow introduction to mathematical topics made them vapid and jejune. Ancient Greeks were enthralled by trigonometry, ancient Egyptians built Pyramids and ancient Chinese built great dams with trigonometry. Calling it practically useless and pedagogically only useful as a prep for calculus is going too far.