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

1,1

Comments

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). The first code in the Mathematica section gives

u = 2.319805307509200010738867057136510870483647988277... ;

v = -1.07556133564118881053529612226074179471679754375... ;

d = 2.557015614241358526013663541906771379699989089781... .

The second code shows (u,v) as the intersection of T and the circle centered at (0,0) with radius d.

The third code shows minimal distance-to-origin points on 16 branches of the tangent function.

Links

Table of n, a(n) for n=1..86.

Formula

u = - sin u sec^3 u.

v = tan u.

d = sqrt(u^2 + v^2).

Example

minimal distance = 2.557015614241358526013663541906771379699989089781...

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 *)

Crossrefs

Cf. A332526, A332527.

Sequence in context: A220426 A117899 A120839 * A196608 A129228 A228587

Adjacent sequences: A332522 A332523 A332524 * A332526 A332527 A332528

Keyword

nonn,cons

Author

Clark Kimberling, Jun 15 2020

Status

approved