A332525 Decimal expansion of the minimal distance between (0,0) and the branch of the graph of y = tan x that passes through (Pi, 0).
2, 5, 5, 7, 0, 1, 5, 6, 1, 4, 2, 4, 1, 3, 5, 8, 5, 2, 6, 0, 1, 3, 6, 6, 3, 5, 4, 1, 9, 0, 6, 7, 7, 1, 3, 7, 9, 6, 9, 9, 9, 8, 9, 0, 8, 9, 7, 8, 1, 2, 2, 8, 7, 7, 1, 8, 6, 6, 8, 9, 0, 4, 7, 4, 9, 1, 3, 7, 0, 4, 0, 1, 1, 5, 5, 6, 7, 8, 6, 6, 2, 0, 0, 5, 1, 2
Offset: 1
Examples
2.557015614241358526013663541906771379699989089781...
Programs
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Mathematica
(* This code computes (x,y) coordinates and the minimal distance. *) x = x /. FindRoot[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2}, WorkingPrecision -> 150] y = Tan[x] d = Sqrt[x^2 + Tan[x]^2] RealDigits[x][[1]] RealDigits[y][[1]] RealDigits[d][[1]] (* Peter J. C. Moses, May 04 2020 *) (* This code shows the two points on the graph of y = tan x and on a circle whose radius is the minimal distance. *) g1 = Plot[Tan[x], {x, -2 \[Pi], 2 \[Pi]}, AspectRatio -> 1]; g2 = Graphics[Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2] &[x /. FindRoot[ FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2}, WorkingPrecision -> 30]]]]; Show[g1, g2] (* Peter J. C. Moses, May 04 2020 *) (* This code shows minimal distance points on 16 branches of the tangent function. *) max = 25; ptX = Table[x /. FindRoot[# == 0, {x, nn}, WorkingPrecision -> 10], {nn, 2, max, Pi}] &[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]]]; Show[Plot[Tan[x], {x, -#, #}, PlotRange -> {-#, #}] &[max], Map[Graphics[{Red, Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2]]}] &, #], Map[Graphics[{PointSize[Large], Point[-{#, Tan[#]}], Point[{0, 0}], Point[{#, Tan[#]}]}] &, #], AspectRatio -> Automatic, ImageSize -> 600] &[ptX] (* Peter J. C. Moses, May 05 2020 *)
Formula
u = - sin u sec^3 u.
v = tan u.
d = sqrt(u^2 + v^2).
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