A Tiresome Cacophony of Superfluous Names - On Circles and Giving Up - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act II

4. On Circles and Giving Up

4.8. A Tiresome Cacophony of Superfluous Names

Let me wind up by pointing out that I do not write these jibes as an opponent of mathematics. I feel that mathematics is important enough not to bury it in symbolic garbage and that those who, with whatever intentions, increase the difficulty of learning mathematics are not taking a serious attitude toward their responsibilities.

—Preston C. Hammer, Standards and Mathematical Terminology

As you might have guessed, our unknown horizontal and vertical pieces show up in the standard textbooks, with predictably unhelpful names. They are called “sine” and “cosine.” Specifically,

H(α) ≡ cos(α)

V(α) ≡ sin(α)

Not content with choosing two archaic and non-mnemonic names for the two simple concepts needed in the above discussion, the standard textbooks then proceed to engage in a Caligula-like bacchanal of terminological overindulgence, in which they conjure up a series of obscure names for simple combinations of V and H, and proceed to make us memorize various quirky behaviors they happen to possess, the entirety of which serves no apparent purpose other than tricking the majority of students into thinking these are genuinely new concepts. Here are some of the unhelpful names found in standard textbooks:

Some older textbooks refer to even more “trig functions,” with names like versine (versed sine), vercosine (versed cosine), haversine (haversed sine), havercosine (haversed cosine), coversine (coversed sine), covercosine (coversed cosine), exsecant, excosecant, hacoversine (or cohaversine), hacovercosine (or cohavercosine). Fortunately, modern textbooks exorcised these latter terms awhile back. They were primarily useful for navigation in the days when large tables of such quantities were a helpful thing to have around. However, in the modern world, where the interesting and important applications of mathematics extend far beyond boats (sorry, boats), and in which we humble primates have managed to build ultra-fast and efficient computing machines for ourselves, it is simply a distraction to bury students under this litany of archaic names. The textbooks abandoned them for that reason.

The quantities still arise regularly, though we hardly realize it when they do, since they are no longer given their own quirky proper-noun-like names. For instance, the now-deceased “trig function” hacovercosin(x) was simply an abbreviation for , and its deceased sibling function vercosin(x) simply stood for 1 + cos(x). Do these two quantities arise in modern mathematics? Absolutely. Do they deserve their own quirky names? Almost certainly not. It’s about time we performed a similar purge of concepts like “secant,” “cosecant,” and “cotangent,” and “tangent,” for the exact same reason.6 They’ve long outlived their use, they deeply confuse the majority of students, and they’re one of many archaic conventions that are killing what might otherwise be a much more widespread interest in our subject. The terms deserve to be put out of their misery, and out of ours.

Although maybe “tangent” can stick around, if it agrees not to make us memorize things about itself.

Okay! For the rest of the book, we’ll reserve the capital letters V and H to stand for what textbooks call “sine” and “cosine.” We chose the letters V and H because they remind us of the words “vertical” and “horizontal,” and the reason we invented the concepts in the first place. However, it is not strictly true that the length to which the abbreviation H(α) refers will always be a horizontal line, and a similar story holds for V (α). This problem occurs with everyday terms too (e.g., the word “left” virtually never means “up,” but it can, if you’re lying on your right side). The rare cases whenH doesn’t refer to the length of a horizontal thing will arise for basically the same reason. However, we’ll always try to make it clear when we encounter such cases, and with that caveat, we’ll forge ahead unapologetically with this much less cumbersome terminology.

4.8.1Picturing All This Another Way

Recall that V (α) and H(α) were abbreviations for V (1, α) and H(1, α). That is, these two symbols stand for the horizontal and vertical lengths spanned by a tilted thing of length 1. Keeping the length fixed while changing the angle will sweep out a circle, as shown in the figure below.

Figure 4.9: Picturing V (α) and H(α), the vertical and horizontal length spanned by a line of length 1 tilted at an angle α relative to the horizontal axis.

Now, if we stare at Figure 4.9, we can see the justification for a different way of visualizing V (α) and H(α): visualizing them as machines depending on α. Staring at Figure 4.9, it shouldn’t be too hard to see several things. First, V (0) = 0 and H(0) = 1. Second, .

Figure 4.10: Picturing V as a machine that eats an angle α and spits out V (α).

Figure 4.11: Picturing H as a machine that eats an angle α and spits out H(α).

Finally, if we increase α by , then we have effectively “spun the clock hand” in Figure 4.9 all the way around, and left it back where it started. This can be stated symbolically by saying that for all α, we have and . Using only these two facts, we can generate two graphs of V (α) and H(α) (see Figures 4.10 and 4.11). To be clear, the above reasoning simply tells us that the graphs of V and H have to somehow periodically wave back and forth, but it doesn’t tell us that the graphs we drew are accurate in every detail. Figures 4.10 and 4.11 are intended to represent only facts we’ve discovered thus far, but the fine-grained details of those graphs aren’t important for our purposes. In both graphs, the horizontal axis shows α, and the vertical axis shows the output of the machines V (α) and H(α).