JBuddenhagen> On Mon, Aug 15, 2011 at 10:04 PM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:

[...]> 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> _______________________________________________>

I guess this must be related to x=4*(sin(Pi/25))^2 being a root of the marvelously simple quintic x^5 - 10*x^4 + 35*x^3 - 50*x^2 + 25*x - 3 + t where t is the golden ratio. ----------- How is that marvelously simple, but not x^5-5*x^2-3? Whose roots are several times less marvelous: Out[1124]= (1/(10^(2/5)))((-1)^((2 #1)/5) (25-5 Sqrt[5]-Sqrt[30 (25-11 Sqrt[5])])^(1/5)+ (-(-1)^(3/5))^#1 (25-5 Sqrt[5]+Sqrt[30 (25-11 Sqrt[5])])^(1/5)- (-(-1)^(1/5))^#1 (-25-5 Sqrt[5]+Sqrt[30 (25+11 Sqrt[5])])^(1/5)+ (-1)^((4 #1)/5) (25+5 Sqrt[5]+Sqrt[30 (25+11 Sqrt[5])])^(1/5))& In[1125]:= Chop@N[#^5-5*#^2-3,69]&/@%/@Range@5 Out[1125]= {0,0,0,0,0} --rwg