A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

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Offset

1,1

Comments

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and a rhombus circumscribed around S, whose vertices lie on coordinate axes.

This constant represents the value of the minimum area of such a rhombus KLMN with vertices K(0,2v), L(-2u,0), M(0,-2v), N(2u,0).

The rhombus touches S at the midpoints of its sides, A(u,v), B(-u,v), C(-u,-v), D(u,-v) which define a rectangle ABCD of the maximum area, inscribed in S, whose sides are parallel to coordinate axes. The constant u can be found as a root of equation x=cot(x) and is known as A069855, and v=cos(u)=u/sqrt(1+u^2).

Links

Table of n, a(n) for n=1..104.

Gleb Koloskov, Geometric illustration

Formula

Equals 8*A069855^2/sqrt(1+A069855^2).

Example

4.4887707055283605403232300252898136708822792436449257365...

Mathematica

N[Minimize[{2 (x+Cot[x])^2 Sin[x], {x>0, x<Pi/2}}, x], 120][[1]]

Prog

(PARI) u=solve(x=0.5, 1, x-cotan(x)); 8*u^2/sqrt(1+u^2)

Crossrefs

Cf. A069855.

Sequence in context: A019674 A264606 A016712 * A333167 A246066 A076359

Adjacent sequences: A346059 A346060 A346061 * A346063 A346064 A346065

Keyword

nonn,cons

Author

Gleb Koloskov, Jul 03 2021

Status

approved