Any angle can be assigned a rational number by dividing a standard angle, say 90deg, into an arbitrary number of segments geometrically, just as points on the line can.

Then any angle can be defined (given a number) by a geometric cut in precisely the same way that any point on the line can.

Any angle in radians is pi times the angle in degrees, and arc length follows accordingly. The rational geometric definirion of angles by rational numbers (degrees), which is more fundamental than radians, was done by the Sumerians 5000 years ago.

With any angle defined geometrically for all x, trigonometric functions are defined geometrically by the standard definitions for all x, and can be treated analytically without series.

And note thatthe standard geometric proof of sinx/x, using geometry and the squeeze theorem, is valid.now

I note in passing that with x and sinx defined analytically without series, SlipEternal’s proof of (x-sinx)/x^{3}< 16, in Challenging limit , post#7, which I am locked out of, is correct, but if I were a teacher I wouldn’t give him credit because he didn’t define sinx analytically without series, which in all fairness, he couldn’t have done without this post.