This may sound dumb, but does such a way exist to mentally (and quickly) determine the values of trigonometric functions such as $\sin(47^\circ)$ and so forth--quickly being a mere matter of seconds? My physics teacher suggested to our class that it is in fact possible, though I see no other ways apart from memorization and the standard methods involving triangles or trig identities (which are decidedly not quick).
I'm on my high school's 'mathletes' team, and though it's unnecessary minutiae I thought that it would be a fun thing to share with my teammates. That, and calculating logarithms mentally, though it appears that the means for doing so has already been answered on this site. (Feel free to comment on the logarithm issue as well, however, if you do have a particularly clever method that you'd be willing to share.)
Parameters: to 2-3 accurate decimal places is ideal. I'm not quite sure how that corresponds %-wise.