# Transform Root objects into Trigonometric expressions

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 @@

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