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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).

%I A332525 #12 Mar 20 2025 04:58:01
%S A332525 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,
%T A332525 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,
%U A332525 0,4,0,1,1,5,5,6,7,8,6,6,2,0,0,5,1,2
%N 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).
%C A332525 Let T be the branch of the graph of y = tan x that passes through (Pi,0). There is a unique point (u,v) on T that is closer to (0,0) than any other point on T. Let d = distance between (u,v) and (0,0).
%C A332525 The first code in the Mathematica section gives
%C A332525   u = 2.319805307509200010738867057136510870483647988277... ;
%C A332525   v = -1.07556133564118881053529612226074179471679754375... ;
%C A332525   d = 2.557015614241358526013663541906771379699989089781... .
%C A332525 The second code shows (u,v) as the intersection of T and the circle centered at (0,0) with radius d.
%C A332525 The third code shows minimal distance-to-origin points on 16 branches of the tangent function.
%F A332525 u = - sin u sec^3 u.
%F A332525 v = tan u.
%F A332525 d = sqrt(u^2 + v^2).
%e A332525 2.557015614241358526013663541906771379699989089781...
%t A332525 (* This code computes (x,y) coordinates and the minimal distance. *)
%t A332525 x = x /. FindRoot[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2},
%t A332525    WorkingPrecision -> 150]
%t A332525 y = Tan[x]
%t A332525 d = Sqrt[x^2 + Tan[x]^2]
%t A332525 RealDigits[x][[1]]
%t A332525 RealDigits[y][[1]]
%t A332525 RealDigits[d][[1]]
%t A332525 (* _Peter J. C. Moses_, May 04 2020 *)
%t A332525 (* This code shows the two points on the graph of y = tan x and on a circle whose radius is the minimal distance. *)
%t A332525 g1 = Plot[Tan[x], {x, -2 \[Pi], 2 \[Pi]}, AspectRatio -> 1];
%t A332525 g2 = Graphics[Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2] &[x /. FindRoot[
%t A332525        FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]] == 0, {x, 2},
%t A332525        WorkingPrecision -> 30]]]];
%t A332525 Show[g1, g2]
%t A332525 (* _Peter J. C. Moses_, May 04 2020 *)
%t A332525 (* This code shows minimal distance points on 16 branches of the tangent function. *)
%t A332525 max = 25;
%t A332525 ptX = Table[x /. FindRoot[# == 0, {x, nn}, WorkingPrecision -> 10], {nn, 2,
%t A332525       max, Pi}] &[FullSimplify[D[Sqrt[x^2 + Tan[x]^2], x]]];
%t A332525 Show[Plot[Tan[x], {x, -#, #}, PlotRange -> {-#, #}] &[max],
%t A332525    Map[Graphics[{Red, Circle[{0, 0}, Sqrt[Tan[#]^2 + #^2]]}] &, #],
%t A332525    Map[Graphics[{PointSize[Large], Point[-{#, Tan[#]}], Point[{0, 0}],
%t A332525         Point[{#, Tan[#]}]}] &, #], AspectRatio -> Automatic,
%t A332525         ImageSize -> 600] &[ptX]
%t A332525 (* _Peter J. C. Moses_, May 05 2020 *)
%Y A332525 Cf. A332526, A332527.
%K A332525 nonn,cons
%O A332525 1,1
%A A332525 _Clark Kimberling_, Jun 15 2020