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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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%I #10 Jul 03 2021 14:54:36

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

%T 6,7,0,8,8,2,2,7,9,2,4,3,6,4,4,9,2,5,7,3,6,5,4,3,6,8,3,2,3,7,4,7,9,9,

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

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

%C 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.

%C 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).

%C 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).

%H Gleb Koloskov, <a href="/A346062/a346062.pdf">Geometric illustration</a>

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

%e 4.4887707055283605403232300252898136708822792436449257365...

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

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

%Y Cf. A069855.

%K nonn,cons

%O 1,1

%A _Gleb Koloskov_, Jul 03 2021