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Consider the Root objects

roots = Table[Root[-1 + 27 #1^2 - 162 #1^4 + 243 #1^6 &, i],{i,1,6}]

These can be expressed in terms trigonometric functions as follows

trig = 
  {-2/3 Cos[Pi/18], -2/3 Cos[5 Pi/18], -2/3 Cos[7 Pi/18], 
    2/3 Cos[7 Pi/18], 2/3 Cos[5 Pi/18], 2/3 Cos[Pi/18]}

This can be checked Numerically, but also exactly using the following expression:

And @@ 
  MapThread[
    PossibleZeroQ[ToNumberField[#1 - #2], Method -> "ExactAlgebraics"] &, 
    {roots, trig}]

True

Is there a way to have Mathematica translate between these two things? I have not been able to make FullSimplify do it. Does anyone know of something like a RootToTrig function?

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3  
FullSimplify[ComplexExpand[ToRadicals[roots]]] –  Daniel Lichtblau Jul 5 at 20:42
    
This ends up throwing N:meprec. –  Matthew Titsworth Jul 5 at 21:11
    
Wrap Quiet around it (it's dealing with zeros that it cannot quite recognize as such). –  Daniel Lichtblau Jul 5 at 21:12
    
Thanks. This works part way in that it gives me trigonometric expressions. However, It still doesn't simplify down to the expressions above. However, that may just be a matter of doing things with FullSimplify. –  Matthew Titsworth Jul 6 at 0:48

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