I don't agree with the perspective taken here; I don't see any reason to think that the definition in terms of triangles is the simplest or most natural, and I suspect the author feels that way just because it's how they were first taught about trigonometrical functions.

Of course I do agree that definitions in terms of power series and differential equations are less natural and require heavier mathematical machinery.

But: the "right-angled triangle" definitions have the severe drawback of only applying to a restricted range of angles, and their relationship to those higher-tech definitions is more indirect than necessary.

Instead, I claim that the One True Definition of the trig functions is: if you start with the point (1,0) and rotate it through an angle t (anticlockwise, which is conventionally the "positive" direction in maths) about (0,0), then the point where it ends up is (cos t, sin t).

This is just as simple, and just as geometrical, as the "triangle" definition. It leads directly to the differential-equation characterization (which in turn leads easily to the power series). For angles between 0 and pi/2, it's obviously equivalent to the "triangle" definition.

Can you get the addition theorems easily from this definition? Yes, and (again) without the severe drawback of only applying when all the angles involved are between 0 and pi/2 as for the "triangle" definition.

Start with a diagram showing the points (0,0), (cos t, 0), (0, sin t), (cos t, sin t). Now rotate the whole thing through an angle u about the origin. Obviously (0,0) stays where it is. (cos t, 0) just goes to cos t times (cos u, sin u), i.e., to (cos t cos u, cos t sin u). If you turn your head through 90 degrees it becomes clear that (0, sin t) similarly goes to (- sin t cos u, sin t sin u). And of course (cos t, sin t) goes to (cos t+u, sin t+u) since we have rotated it through an angle t and then through an angle u.

And now we're done, because the "vector addition" property the original diagram had remains true after rotation, so adding (cos t cos u, cos t sin u) to (- sin t cos u, sin t sin u) has to give you (cos t+u, sin t+u). And that's exactly the addition theorems.

(That may be hard to follow with no diagrams, but I think it's easier to follow than the triangle-stacking proof would be without diagrams, and with diagrams everything is pretty transparent.)

The right-angled-triangle definitions are traditional because historically trigonometry came before coordinates. But now we have coordinates and generally learn about them before we learn trigonometry, and at that point the point-on-the-unit-circle definitions are simpler, more general, and better suited for proving other things.