sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?
If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles.
It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
By having a right triangle?
The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles.
Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps.
But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that.
(Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.)
(I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
That said, if you have enough Calculus to define how to measure the arclength of a segment of the circle, you can quickly prove that sin and cos in radians exist, have a nice power series, and so on.
It is like x^y with x positive. We can manually define it every rational y. But the easiest way to get a rigorous and straightforward definition is to prove the algebraic properties of the integral of 1/x, use that to define the logarithm, define its inverse function to be the exponential, prove its algebraic properties, then define x^y as e^(y*log(x)). And it all just works.
But even a disembodied being of pure reason might eventually discover continuity via logic->topology.
Of course that assumes that arclength is well-defined. The standard approach to which is, of course, Calculus.
If you don't have calculus, you don't have anything like a delta-epsilon proof of continuity. But without calculus, you also don't know that you need it. So you just assume (correctly) that you can interpolate, and it works just like you expect, and life goes on.
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.
What other foundation or learning pathway do you see trig serving as? Somebody else mentioned that trig serves use by teaching students that calculus has rich applications. So then I question, what kind of applications are students learning in trig? And if students are to learn rich examples of calculus applications, then why not statistics, which is also relevant to the bio / social sciences? Also, couldn't we mash trig inside calculus?
Then I take physics, and I find a whole bunch of other applications. I take calculus, and I find a bunch more uses. I take mechanics, and I find a bunch more. But it is not the job of trig to teach me those applications (though hints would be useful). It's not trig's job to teach me physics - that's a job for physics. But I need trig as a foundation.
I'm not sure that I answered your question, though...
I would say a course on trigonometry usually covers (my experience): trigonometric functions exact value of them for the angles 30, 45, 60, 90 ... degrees Trigonometric formulas for the sum and difference of angles. A formuka for the double and the half angle. Law of sine and law of cosine Lots of relations derived from the Pythagoras theorem (sin^2+cos^=1) how to solve trigonometric equations
With all this, you are equipped to completely determine a triangle, knowing some of its and the length of some its sides. As as application, I was taught, how to measure heights and distances provided you can measure angles.
Thus, without trigonometry, it would be fairly hard to take a course on analytic geometry.
Now, how would the course be enhanced by introducing sine as the solution of an ODE?
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)...
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?
"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?" (http://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus#Py...)
Well, maybe not that typical, but it is an example without any periodicity in sight.