I do not know, if ever posted this. But it seems long forgotten knowledge, like so much of elementary geometry. Students coming to university, even math students, usually only know these measures as function curves, or know how to perform calculus with them. But the geometric idea seems to be lost somewhere along the way.

By the way, if you need some exercise, express all these items in terms of the sin! You can use rational geometry, if you like.

To look at the trigonometric sizes as functions is a different story. E.g., sine and cosine easily make sense for all angles, since they are simply the coordinates of a point

\((\cos(x),\sin(x)) \)

on the unit circle. Here, x is the angle as in the image above. But if we define the angle in radians, it is simply the curve length the point has run through on its travel around the unit circle, starting from the point (0,1).

Asking for inverse functions is still another story. The geometry of the tangens can be seen from the following ímage.

Seen like this tangens becomes a function

\(\tan : [-\pi/2,\pi/2] \to \mathbb{R} \)

and has an inverse function in this interval.

As you see, math concepts depend on the things you want to achieve. There is never a canonical definition. You have to involve both parts of your brain, the geometrical and the logical.