From http://mathworld.wolfram.com/TrigonometryAnglesPi11.html : [...] hence [image: sin(pi/11)], can be expressed in terms of radicals (of complex numbers). The explicit expression is quite complicated, but can be generated in *Mathematica <http://www.wolfram.com/products/mathematica/>*using Developer`TrigToRadicals<http://reference.wolfram.com/mathematica/Developer/ref/TrigToRadicals.html> [Sin <http://reference.wolfram.com/mathematica/ref/Sin.html>[Pi<http://reference.wolfram.com/mathematica/ref/Pi.html> /11]].

However, Developer`TrigToRadicals[Sin[\[Pi]/11]] Developer`TrigToRadicals::obs: Developer`TrigToRadicals has been superseded by ToRadicals, and is now obsolete. It will not be included in future versions of Mathematica. >> and then disgorges 13 pages(!) of 5th roots. Trying In[1058]:= ToRadicals[Sin[Pi/11]] Out[1058]= (-(1/2))*(-1)^(9/22)*(-1 + (-1)^(2/11)) Foo! I wanted 5th roots. Just far fewer. Like this: Out[1097]= -(1/5) + (1/(5 2^(2/5)))11^(1/5) E^((2 I \[Pi] #1)/ 5) ((-89 - 25 Sqrt[5] - 5 I Sqrt[2 (205 - 89 Sqrt[5])])^(1/5) - (-1)^( 3/5) (-89 + 25 Sqrt[5] - 5 I Sqrt[2 (205 + 89 Sqrt[5])])^(1/5) E^((2 I \[Pi] #1)/5) + (-1)^( 2/5) (-89 + 25 Sqrt[5] + 5 I Sqrt[2 (205 + 89 Sqrt[5])])^(1/5) E^((4 I \[Pi] #1)/5) + (-89 - 25 Sqrt[5] + 5 I Sqrt[2 (205 - 89 Sqrt[5])])^(1/5) E^((6 I \[Pi] #1)/5)) & In[1099]:= Rationalize[ArcCos[Chop[(%1097 /@ Range[5])/2.]]/\[Pi]] Out[1099]= {2/11, 10/11, 6/11, 8/11, 4/11} In[1100]:= Rationalize[ArcCos[Chop[(%1097 /@ Range[5])/-2.]]/\[Pi]] Out[1100]= {9/11, 1/11, 5/11, 3/11, 7/11} In[1101]:= Rationalize[ArcSin[Chop[(%1097 /@ Range[5])/2.]]/\[Pi]] Out[1101]= {7/22, -9/22, -1/22, -5/22, 3/22} I.e., the pi/11 article could give all the sins and coses with just %1097, and all the cscs and secs with a similar expression. One wonders how much smaller could be the expression for Sin[pi/23], whose article says it takes .5GB! This may have discouraged trying pi/25, which http://mathworld.wolfram.com/TrigonometryAngles.html skips, yet Simplify[Developer`TrigToRadicals[Sin[\[Pi]/25]]] (after the death threat) produces this marvelously simple quintic surd -((I (-(-1 - Sqrt[5] - I Sqrt[10 - 2 Sqrt[5]])^( 6/5) + (-1 - Sqrt[5] + I Sqrt[10 - 2 Sqrt[5]])^(6/5)))/ (8 2^(2/5))) And ToRadicals doesn't.-( --rwg