Just for grins, switching to Maple, sin(Pi/34) = (1/8)*sqrt(2)*sqrt(15+sqrt(17)-sqrt(34-2*sqrt(17))-sqrt(68+12*sqrt(17)+2*sqrt(34-2*sqrt(17))-2*sqrt(578-34*sqrt(17))+16*sqrt(34+2*sqrt(17)))) and the minimal polynomial Bill Gosper gives factors over Q(sqrt(17)). The quartic factor 16*x^4-(4*sqrt(17)-4)*x^3-(2*sqrt(17)+6)*x^2-(-2*sqrt(17)-4)*x-1 has sin(Pi/34) as a root. The other roots are sin(9*Pi/34), sin(13*Pi/34), sin(15*Pi/34) and -sin(19*Pi/34). On Sat, Oct 19, 2019 at 12:47 AM Bill Gosper <billgosper@gmail.com> wrote:
Brad, there's a really interesting answer. E.g., In[68]:= Sin[π/34] // FunctionExpand
Out[68]= 1/(4 Sqrt[2/( 15 + Sqrt[17] - Sqrt[2 (17 - Sqrt[17])] - Sqrt[ 2 (34 + 6 Sqrt[17] + Sqrt[2 (17 - Sqrt[17])] - Sqrt[ 34 (17 - Sqrt[17])] + 8 Sqrt[2 (17 + Sqrt[17])])])])
(Rather cumbrous, Now undo it.)
In[69]:= ArcSin@% // FullSimplify // Timing
Out[69]= {0.312125, π/34}
How could it take .3 sec to look that up? Alternatively, how could it get π/34 so quickly? Or at all?
Here Mathematica recognizes Macsyma's nicer rendition: In[70]:= ArcSin[(-\[Sqrt](\[Sqrt]17 + 3) \[Sqrt](4 \[Sqrt]17 - 2 \[Sqrt](34 - 2 \[Sqrt]17)) + \[Sqrt](34 - 2 \[Sqrt]17) + \[Sqrt]17 - 1)/16] // FullSimplify // Timing
Out[70]= {0.948516, π/34}
even though it wasn't able to simplify its version to Macsyma's. So where did it get the notion of π/34? It guessed it numerically! It is then fairly routine to show that the two surds are identical, or that Macsyma's satisfies In[72]:= MinimalPolynomial[Sin[π/34], x] // Timing
Out[72]= {0.002874, 1 - 8 x - 40 x^2 + 80 x^3 + 240 x^4 - 192 x^5 - 448 x^6 + 128 x^7 + 256 x^8}
At least this is how Macsyma did it. Without even requiring a ratsimp or radcan. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun