Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act II
4. On Circles and Giving Up
4.7. Molière Is Dead! Long Live Molière!
At this point, we’ve got no more ideas. We’ve failed to solve the problem, although we did figure out that we didn’t need both slots in V and H. We don’t have the foggiest idea of how to calculate specific numbers for H(α) or V (α) in general, but for very specific angles we might be able to think of a trick. For example, when α = 0, we’ve got a completely horizontal line, so H(0) = 1 and V (0) = 0. Similarly, when α is a quarter of a full turn, we’ve got 
, because of the strange convention for measuring angles such that a full turn counts as 
. But then for , we’ve got a completely vertical line, so 
and 
. If α is a “45-degree angle,” or an eighth of a full turn, so that 
, then the horizontal and vertical parts are the same length, so we’ve got V = H. But then, since H and V were defined to be the horizontal and vertical lengths of a tilted thing of length 1, we can use the formula for shortcut distances together with V = H to get 12 = V2 + H2 = 2V2, which tells us that
If we think a bit harder, we might be able to come up with a trick that lets us figure out what V and H are for a few more specific angles, but we have no good reason to do this. Even if we did, it would still be a far cry from solving the problem we set out to solve.
We’re completely stuck. We failed. And so we pull the Molière trick yet again. Just like before, having failed to solve a seemingly simple problem involving circles, we simply give up, and act as if the names V and H that we gave to the unknown solutions were the solutions themselves! Didn’t know we could do that? Sure we can. That’s what every introductory trigonometry book does!5
By the way, the name “trigonometry” is misleading. The point is not to study triangles. The point is to break up tilted things into horizontal and vertical pieces. As we saw in Figure 4.7, triangles arise by accident, as a side effect of this. The reason they happen to be “right triangles” is just because horizontal and vertical things are perpendicular. Trigonometry is a non-subject.
Of course, they never tell us that’s what they’re doing, so we usually end up blaming ourselves for not understanding a problem that the books and teachers manifestly do not solve. It’s not until we’re well into the world of calculus that we finally have the tools to solve this problem. At that point, we can at last write down an explicit description of these machines V and H that are so mysterious in the absence of calculus; a description just as explicit as the descriptions of our plus-times machines; a description that lets us calculate V (α) and H(α) for any angle, to whatever accuracy we want. We’ll arrive at that point in the next interlude, when we invent the nostalgia device. Stay tuned!