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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). 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified April 11 02:27 EDT 2021. Contains 342886 sequences. (Running on oeis4.)