Re: Stupid Mma Tricks

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Re: Stupid Mma Tricks*From*: keiper (Jerry Keiper)*Date*: Sat, 12 Mar 1994 15:25:14 -0600

> You also get exact answer for other multiples of pi/12 and other trig > functions. However sin(pi/8), sin(pi/24), etc. received no such special > attention -- they remain unevaluated. Apparently somebody thinks > Pi/12 ( = 15 degrees) is special. > Then it struck me. What if all exact trig values are calculated > through some kinky algorithm that deals strictly with multiples > of pi/12, so that e.g. when you find sin(pi/3), it is handled > as sin(4 pi/12)? Perhaps Mathematica calculates the fourth power > of the 2x2 rotation matrix for an angle of pi/12, then extracts > the values of sin(pi/3) and cos(pi/3) from that matrix? > Would somebody please confirm or deny this ugly rumor? It is generally accepted that Pi/6 and Pi/4 are sufficiently special that the trig functions should evaluate at them. Because nothing is calculated (the values are stored in a table) it was convenient to have a table with no holes in it. GCD[1/6, 1/4] is 1/2 so the tables are set up in increments of Pi/12. The argument of the function is multiplied by 12/Pi (and reduced by symmetries, etc) to give the index into the tables. Multiples of Pi/8 and Pi/5 don't fit into the tables. Of course we could set it up in increments of Pi/120 to include these values, too ... Jerry B. Keiper keiper at wri.com Wolfram Research, Inc.