“Move x-sub-one to the left, then square both sides…”. Ruben stepped carefully from expression to equivalent expression. “…wait, was theta-naught positive, or negative?”
The problem seemed simple enough, just a system of four linear equations. Well linear-ish, three of the four unknowns were angles, and they were in trig functions, but otherwise totally linear. Solving for the first variable was textbook, just add two equations together to cancel out an unknown, whip out a couple trig identities, and there you have it. The second fell out by the same process, no sweat. Rube felt very satisfied, almost cocky.
Then came the Russian winter. He had to substitute the two variables back into the first equation and solve for the third unknown. The equation said, “I’ve had enough”, and dug in for a siege. At first Ruben tried to blaze forward, like he had in the beginning, exploiting two and three identities per step. He wasn’t going to be delayed by a little ugly math. But the expressions kept getting uglier, terms combined rather than canceling; he must have made a mistake.
Rube started backtracking. Two steps above, he found a positive where there should have been a negative. Then, in another step, he had forgotten to double the product when squaring a sum. In still another he had canceled cosines that had different angles. The errors were adding up too fast. As he tried to unravel the consequences of each prior mistake he created more errors. Logical threads wound knots, constricting his mind. There was no other way; the problem had to be restarted.
Rube made his retreat, and made preparations. Graphite was stocked, paper was stacked. This time he would do it like he was taught. This time there would be no mental warfare. This time the action would only take place on paper.
The siege has been long, but Rube’s patience has been unwavering. One step, then another. Add to both sides, then expand the sine of a sum of angles. Victory is not yet had, but it is assured.