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
Gleb Koloskov, Geometric illustration
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
KEYWORD
nonn,cons
AUTHOR
Gleb Koloskov, Jul 03 2021
STATUS
approved