I am finally making headway in this arcane section of trigonometry. I am far from the ocean breezes, distant misty landmarks, and lengthening shadows of beginning trigonometry, and into a world of rigorous abstractions in which there are only two dimensions. I began to make sense out of these trig identities and the problems when I realized that these are built just like those polynomial problems of simplification and manipulation which concerned me back in 2001.

I well remember my polynomial passage. I spent endless hours with a² and b² and a³ and b³ and ab and ab² and all those other letters and exponents and adding and multiplication and division and factoring and canceling out and then multiplying them back together. Here in Trigonometry the "a" and the "b" are replaced by "sinA" and "cosA" and other such things. And there are some different rules in this business, such as sin²A + cos²A = 1. But the process of simplification and mooshing the expression around is basically the same as polynomial-pushing, or so it seems to me.

I did polynomials in October and November of 2001, when a large part of New York City was a smoking hole in the ground, and a large chunk of the Pentagon (7 miles from my home) was a burnt-out ruin. The news in the Washington Post isn't much better now; it just isn't our turn this time. Is math an escape from the horrors of the world? It is a cold but satisfying comfort, knowing that these trigonometric relations would be true (at least in two dimensions) even if humanity disappeared from the universe.

All right, I admit it. I look at the answers in the book. Schaum's Outlines (to be precise, the authors of Schaum's) are only too happy to show me how a problem was done. This is how I'm learning what to do. In the absence of a real live teacher, the book suffices. No human being could be as patient or as helpful as the good books; they are there morning, noon, and deep into the night. They rarely make mistakes, and they never complain. Until a "virtual professor," an artificial intelligence with a realistic and responsive "human" interface is created, the books and the websites will do the job. They are unmoved by disaster, tragedy, misery, or sadness — like mathematics itself.

Posted at 2:02 am | link