A332500 -id:A332500 - cp's OEIS frontend

A332526 Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments.

2, 3, 7, 5, 0, 6, 9, 1, 4, 6, 0, 4, 0, 1, 7, 6, 3, 4, 9, 4, 3, 9, 8, 5, 1, 5, 5, 8, 7, 7, 8, 9, 8, 2, 4, 8, 7, 8, 6, 6, 2, 6, 7, 8, 0, 6, 5, 0, 8, 8, 4, 1, 7, 9, 2, 9, 2, 6, 9, 8, 5, 6, 4, 5, 9, 7, 5, 4, 8, 6, 6, 7, 0, 2, 9, 6, 9, 1, 3, 1, 6, 3, 3, 4, 1, 1

Offset: 1

Clark Kimberling, Jun 15 2020

Let T0 and T1 be the branches of the graph of y = tan x that passes through (0,0,) and (Pi,0), respectively. There exist points P = (p,q) on T0 and U = (u,v) on T1 such that the distance between P and U is the minimal distance, d, between points on T0 and T1.
u = 2.549082584017596768984130292562154758705824602711...
v = -0.67319711901285205370684801604861382107848678888...
p = Pi - u
q = - v
d = 2.375069146040176349439851558778982487866267806508...

			minimal distance = 2.375069146040176349439851558778982487866267806508...

Cf. A332500, A332525, A332527.

min = Quiet[FindMinimum[Sqrt[(#[[1]][[1]] - #[[2]][[1]])^2 + (#[[1]][[2]] - \
#[[2]][[2]])^2] &[{{#, Tan[#]} &[x /. FindRoot[# Cos[#]^2 - x Cos[#]^2 + Tan[#] == Tan[x], {x, 0}, WorkingPrecision -> 500]], {#, Tan[#]} &[#]} &[y]], {y, 2}, WorkingPrecision -> 100]]
Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 + Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, 0, Pi}, AspectRatio -> Automatic, ImageSize -> 300, PlotRange -> {-2, 2}], Graphics[{PointSize[Large], Point[{Pi/2, 0}], Point[{#, Tan[#]}], Point[{Pi - #, -Tan[#]}]}]] &[y /. min[[2]][[1]]]
(* Peter J. C. Moses, May 06 2020 *)

A332522 Decimal expansion of the number u such that the line normal to the graph of y = sin x at (u, sin u) passes through (Pi/4,0).

Original entry on oeis.org

4, 1, 5, 8, 5, 5, 5, 9, 6, 7, 8, 9, 8, 6, 7, 9, 8, 8, 7, 8, 8, 0, 0, 4, 8, 0, 1, 9, 8, 2, 9, 3, 9, 0, 1, 9, 0, 4, 2, 5, 8, 6, 4, 7, 0, 3, 9, 3, 3, 9, 7, 7, 2, 2, 7, 6, 0, 8, 9, 5, 8, 3, 0, 1, 9, 0, 3, 1, 1, 7, 7, 4, 1, 2, 9, 7, 7, 8, 4, 5, 2, 3, 5, 7, 3, 2

Offset: 0

Clark Kimberling, May 05 2020

			u = 0.4158555967898679887880048019...

Cf. A332500, A332507.

u = u /. FindRoot[0 == (-Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 0}, WorkingPrecision ->  150]
RealDigits[u][[1]]

cp's OEIS Frontend

A332526 Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments.

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