I thought maybe it was about time I post something actually useful here. It occurred to me that the method I use to memorize my trig tables is something that my students never seem to have seen, so maybe others haven't either. I don't claim it's brilliant, but I was pleased to notice it.
The first thing you have to remember is the angles that you are supposed to remember: 0, π/6, π/4, π/3, π/2 (in that order). Now, lets start with the values for sin. Going from least to greatest (in the angles), the values for sin are (√0)/2, (√1)/2, (√2)/2, (√3)/2, and (√4)/2. Writing them this way, it is easy to see the pattern. They're all (√n)/2, where the n goes from 0 to 4. Of course, sin(0), sin(π/6), and sin(π/2) simplify to 0, 1/2, and 1, respectively. To get the values for cos, reverse the process. The way I do it is to go backwards in the angles (π/2 to 0), but use the same ordering for the values.