[replicant]

It is about triangles. The idea that sine is the solution of this ODE for f(0)=0 and f'(0)=1 is quite modern.

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?