In trigonometry, I'm now working on angle addition and subtraction formulas, double-angle formulas, and half-angle formulas, along with their various identities and problems involving their interrelations. This is classic trig, and if I were a True Mathematician I'd probably find it entertaining just for the endless recombination of pattern and re-statement. I'm not true enough, then, because I find it truly tedious. Not only that, I keep wondering what all this stuff is for.

OK, I know, I'm not supposed to ask "what things are for" in mathematics, that's gauche. As the math mantra goes, "It will be revealed to me as my studies progress." But I can't help asking this simpleminded question: what's the point of finding the sine/cosine of the half-angle using elaborate half-angle equivalency formulas, when you already know the whole angle and could just look up the results in the table at the back of the book, let alone using your pocket calculator? There must be some importance to these calculations and formulas, otherwise the book wouldn't spend so much time and space on them. I just don't know what they are yet.

Many years ago, a certain group of people (science fiction fans, I admit it), enjoyed watching a short film called "Closet Cases of the Nerd Kind," made in 1980 as a parody of Spielberg's CLOSE ENCOUNTERS OF THE THIRD KIND (1977). I have seen it more than once, with much hilarity; in fact it is one of the funniest short films I've ever seen. (You can find a ridiculously expensive "official" video of the film and a short description at this site.)

In the film, the character parodying Richard Dreyfuss repeatedly sculpts mysterious forms with gushy substances like whipped cream, mashed potatoes, and shaving cream. He is receiving mystical visions of alien arrivals, but he doesn't know what they are yet. As he contemplates yet another skwushy disc-shaped sculpture in his hand, he almost reaches a moment of enlightenment, and he says, "This MEANS something." And then, before he says more, his hand automatically slaps the skwushy pie into his face.

This is how I feel with these trigonometric formulas. These malleable, squeezable sets of numbers squish around on my math paper, Pythagorizing themselves and inverting themselves and sometimes melting away to a single expression. They have long since ceased to resemble hard-edged triangles or configurations of ships and lighthouses. Somewhere out there in math and physicsland, these things are important. But right now, I'm there with the Nerd Kind, contemplating the whipped-cream form before me, trying to solve problem after problem ("Prove that cos2x = cos⁴x — sin⁴x.") The ever-helpful Schaum's suggests using the "difference of squares" to deconstruct this, which melts it right away. But then there's another problem, and then another. I stare at it. This means something. The mathematical pie in the face is coming, and I know that it is calculus.

Posted at 2:47 am | link