I'm sure Dave's solution is perfectly right, and I don't want to think about it too much since mystery hunt is frying my brain. Anyway, I did wonder whether there isn't some decent way to estimate t - sin t = pi / 4 by hand:

the taylor series for sin is t - t^3/3! + t^5/5! - ... Therefore our equation is, pi/4 = t^3/6 - O(t^5). Hence we'd expect that t is roughly the cube root of 3 pi / 2. This gives you 1.7. To check, 1.7 - sin 1.7 = .71, while pi/4 = .79, so it isn't that good, but it also isn't that bad. So, if you happen to be stuck with a table of sines but not a graphing calculator this isn't a bad way to estimate it. Although once you have a reasonable idea around what the answer is you could get it better with guess and check with this table, so its not that useful.

This exercise did do a very good job of calming me down to go to sleep though.