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.<p>Of course I <i>do</i> agree that definitions in terms of power series and differential equations are less natural and require heavier mathematical machinery.<p>But: the "right-angled triangle" definitions have the <i>severe</i> drawback of only applying to a restricted range of angles, and their relationship to those higher-tech definitions is more indirect than necessary.<p>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).<p>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.<p>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.<p>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.<p>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.<p>(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.)<p>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.