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URL: https://oeis.org/A332525

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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
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.
FORMULA
u = - sin u sec^3 u.
v = tan u.
d = sqrt(u^2 + v^2).
EXAMPLE
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
Sequence in context: A220426 A117899 A120839 * A196608 A129228 A228587
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 15 2020
STATUS
approved