A351582 Decimal expansion of the root of cot(Pi/(s+1)) - csc(Pi/s).

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

Offset: 1

Author

Robert B Fowler, Feb 14 2022

Keywords

• cons
• nonn

Comments

For regular unit-sided polygons with number of sides s >= 3, the s-gon fits inside the (s+1)-gon, and hence inside any t-gon where t > s. For s = 3 and s = 4, this is verified by diagram. For s >= 5, it is verified by observing that the s-gon's circumcircle is smaller than the (s+1)-gon's incircle. The difference of the two circles' radii is negative for s <= 4 and positive for s >= 5, and changes sign at non-integer value s = 4.49547...

Diagrams demonstrating this property of regular s-gons are interesting (see links).

Examples

			4.4954747887528...
		

Links

• Robert B Fowler, Diagram of Nested Unit-sided Regular Polygons (s=3 to s=12)
• Luxor, Diagram of Concentric Unit-sided Polygons (s=3 to s=20). See diagrams near the start and near the end of the article. The triangle, square and pentagon intersect.

Programs

Maple

Digits:= 120:
fsolve(cot(Pi/(s+1))-csc(Pi/s),s);  # Alois P. Heinz, Feb 16 2022

Mathematica

RealDigits[s /. FindRoot[Cot[Pi/(s + 1)] == Csc[Pi/s], {s, 4}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 14 2022 *)

PARI

solve(s=4, 5, cotan(Pi/(s+1)) - 1/sin(Pi/s)) \\ Michel Marcus, Feb 14 2022

Formula

For integer values of s >= 3:

c(s) = circumcircle radius of unit-sided regular s-gon = csc(Pi/s) / 2,

i(s) = incircle radius of unit-sided regular s-gon = cot(Pi/s) / 2,

d(s) = i(s+1) - c(s),

d(s) <= 0 for s <= 4, d(s) > 0 for s >= 5.

For real values of s:

d(1) = -infinity,

d'(s) > 0 for s > 1,

d(s) = 0 for s = 4.4954747887528...