I expected to be done with basic trig by now, but here I am, still memorizing formulas and doing trigonometric identity problems, for the third month in a row. I can now incorporate angle addition and subtraction, double-angle, and half-angle formulations into these workings. I assume that these will reappear someday in my studies, and I will say, while contemplating some mysterious oscillation or repeating phenomenon, wow! that must be the result of sin(A+B) or sin2A or something of that sort that I learned back in 2004.

I feel, pedantic and dull as it sounds, that I must go through all of these identities and learn them and learn how to work with them, lest I be caught short in the middle of some calculation somewhere way in the future. Will there be a TEST? Who will test me? One of my wry and imaginative friends has suggested an eschatological test. That is, at the end of time (or the end of MY time), I will face Jesus at the last judgement (or the judgement of my individual soul). And instead of the Biblical injunctions to do well by one's fellow human beings (see Matthew 25:31-46) asked of people at the Last Judgement, Jesus will approach me and say… what is the formula for the addition of the tangents of angle A and B? And what is the relationship of tangent and secant? And if I say, "Uh, I don't remember," I will fall into the lake of fire.

Well, maybe not so dramatic. And yet, Jesus according to Christian theology is a Person of a Trinity, which has three entities, thus a triangle. So Jesus Himself has a sine and a cosine, in heavenly trigonometry. In some artworks you can actually see the triangle behind Jesus' halo. If this is an equilateral triangle, which it usually is when the Trinity is symbolized in art, then Jesus is 60 degrees, thus making his sine the square root of three over 2 and his cosine one half (0.5). That means that Jesus' sine is irrational! We behold the sine of the divine, and it is irrational! Fortunately his cosine is not. He does, after all, have two natures in one divine Person.

My apologies to my Atheist or non-Christian readers for these after-midnight riffs. I am sometimes driven to distracted fantasy when facing yet another round of these trigonometric identity problems. I'm not done yet. I need to go through the Product of Sines and Cosines and the Sum and Difference of Sines and Cosines, which is the next short section in the red-backed Schaum's Outlines. After that, things might start moving, or at least oscillating, as I will be studying graphs of trigonometric functions, trigonometric equations and their graphs, and waves.

The identity problems are teaching me a way to look at mathematics which I could describe either as the "Russian nesting doll" model, or the "Chinese box" model, or perhaps the "Japanese Transformer Robot" model. These Orientalizing (OK, Russia is "Eastern" Europe) concepts have to do with nested and transforming entities. I followed an example problem in the Schaum's text which went on for a whole page in my notes, as one expression was substituted with another, which was then re-arranged with its neighbors to create yet another expression which could be substituted. And then that new entity was itself part of yet another trigonometric statement which could be re-stated to match its neighbors, thus working another transformation which eventually unfolded or infolded into the desired configuration. Entities within entities, hidden formulas I have to look for, substitutions I must not fail to make; if it's not an interlocking puzzle box, it's a labyrinth to get lost in. I must know my landmarks, hence all this going back and forth in which I have spent my winter.

For those mathematical experts who read this Weblog (I hope you're out there), this is the Orientalizing labyrinth which I was fortunate enough to be guided through: "When A + B + C = 180 degrees, show that sin2A + sin 2B + sin 2C = 4 sinAsinBsinC." Easy for you, perhaps… I hope the Last Judgement doesn't come soon.

Posted at 1:34 am | link