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Stupid Mma Tricks


One of my students yesterday used Mathematica to evaluate sin(pi/12), and guess  
what came out!

In[1]:= Sin[Pi/12]
Out[1]=
-1 + Sqrt[3]
------------
 2 Sqrt[2]

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?

Richard Mercer





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