Date: 2019-10-19 09:23 From: R B <bossipm@outlook.com> Some googling gave me https://nanopdf.com/download/sines-and-cosines-for-fractions-of_pdf from which Sin[π/34] can be derived simply. Arie Bos "So, sin π/n and cos π/n can be found for n = 2k ×3p ×5q ×17r ×257s ×65537t where k is a positive or zero integer and the indices p, q, r, s, t are all either zero or 1." RubbEESSHimo! In[90]:= #@Sin[π/7] & /@ {ToRadicals, Developer`TrigToRadicals} During evaluation of In[90]:= Developer`TrigToRadicals::obs: Developer`TrigToRadicals has been superseded by ToRadicals, [I believe this to be completely false.] and is now obsolete. It will not be included in future versions of the Wolfram Language. [And I pray this will not happen.] Out[90]= {-(1/2) (-1)^(5/14) (-1 + (-1)^(2/7)), -(1/2) I (1/3 (1/ 2 (1 - I Sqrt[7]) - ((-1 + I Sqrt[3]) (1/2 (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1/2 (-1 + I Sqrt[3]) + 1/4 (-1 + I Sqrt[3])^2)))/( 2 (6 + 3/4 (-1 + I Sqrt[3]) (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) - 1/4 (-1 + I Sqrt[3])^2 (6 + 3/4 (-1 + I Sqrt[3]) (-1 + I Sqrt[7]) + 1/2 (-1 - I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) + 1/3 (1/2 (-1 - I Sqrt[7]) + ((-1 + I Sqrt[3])^2 (1/2 (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1/2 (-1 + I Sqrt[3]) + 1/4 (-1 + I Sqrt[3])^2)))/( 4 (6 + 3/4 (-1 + I Sqrt[3]) (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)) + 1/2 (-1 + I Sqrt[3]) (6 + 3/4 (-1 + I Sqrt[3]) (-1 - I Sqrt[7]) + 1/2 (-1 + I Sqrt[7]) (1 + 3/4 (-1 + I Sqrt[3])^2))^(1/3)))} In[325]:= MinimalPolynomial@% Out[325]= {-7 + 56 #1^2 - 112 #1^4 + 64 #1^6 &, -7 + 56 #1^2 - 112 #1^4 + 64 #1^6 &} In[328]:= %%%[[1]]@Sin[π/7] // FullSimplify Out[328]= 0 How can that bullbleep page remain up? —rwg

-----Oorspronkelijk bericht----- Van: math-fun <math-fun-bounces@mailman.xmission.com> Namens Bill Gosper Verzonden: zaterdag 19 oktober 2019 07:46 Aan: math-fun@mailman.xmission.com CC: Wolfram Technical Support <support@wolfram.com> Onderwerp: [math-fun] "LHS Sometimes returns precision errors? What the bleep is it doing numerics for an identity map?" Urgentie: Hoog

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