On the other hand, often our tools have somewhat peculiar forms. For example, “trigonometry” as commonly taught in high schools and used in science/engineering/mathematics uses bizarre names, terrible notation conventions, and a giant pile of unmotivated formulas to memorize.

Teaching a bit of history alongside helps students understand why it takes that particular form.

The word “sine” comes from a weird Latinization of an Indian word for “half a bowstring”. Draw a picture of a circle with a vertically oriented chord (the word “chord” also implies a bowstring), and a student will have a much easier time remembering what the sine is. Likewise tangent (Latin for touching) and secant (Latin for cutting) make more sense if you think about the meanings of the words. Cosine means the sine of the complementary angle, etc.

The reason we call inverses “arcsine”, etc. is because originally these were written as quasi-sentences, and the concept of a mathematical function was not well developed. So sin⁻¹ x would be expressed as something like: arc (sin. = x). That is, the length of the arc whose sine is x. This form was cumbersome so later got shortened to arcsin x.

The origins of trigonometry are in astronomical measurement, which is why we have 360° in a circle (each degree is roughly one day of movement (365 days/year), rounded to a nearby highly composite number), and come from the Sumerian/Babylonian numerical tradition which used a base sixty number system. Hence “first minutes”, “second minutes”, “third minutes”, etc. of a degree. “Minute” (Latin for small) implies 1/60 of the larger unit.

The reason trigonometry focuses on learning a big pile of formulas is because before the era of electronic calculators, people needed to do all computations by hand, or by interpolating in pre-computed lookup tables. The goal of “trigonometry” is to take a given problem and convert it to a form with the easiest hand computation and the fewest table lookups possible, so that the mechanical work can be handed off to a team of human computers who can go through the laborious arithmetic. Memorizing trigonometry formulas is a way to cut the work done by the human computers to a small fraction of what it might take for the original problem as posed.

Trigonometry was important in science/engineering because until recently the abstract vector concept and idea of combining simple single-number parts into “complexes” were not well developed. People solved problems by breaking them into coordinates and discrete lengths and angles. Solving triangles and converting between polar/cartesian coordinates were important steps in almost any 2-dimensional problem.

If we really wanted to avoid the “ontogeny recapitulates phylogeny” model, we would scrap the current form of trigonometry (certainly not spend 4+ months exclusively focusing on it) and set the high-level ideas on a more logical foundation which was easier to learn and reason about, ditching the parts now anachronistic in an electronic computer age. We would give students harder problems to solve and fewer formulas to memorize. But that could leave students unfamiliar with the existing language commonly used in the existing literature, so to some extent we’re stuck by our history. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf