A384805 Consider triangles ABC satisfying (sin A)^2 + (sin B)^2 = sin C. Sequence gives A_0 (in radians), the maximum A such that more than one triple (A,B,C) is possible.

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

Offset

0,1

Comments

In other words, maximum A in (0,Pi) such that (sin A)^2 + (sin B)^2 = sin(A+B) has more than one solutions B in (0,Pi-A).

Number of triples (A,B,C), A + B + C = Pi, satisfying (sin A)^2 + (sin B)^2 = sin C:

If Pi/2 < A < Pi, then there exist exactly one solution (A,B,C);

If A_0 < A < Pi/2, then there exist exactly one solution B = Pi/2 - A, C = Pi/2;

If A = A_0, then there exist exactly two solutions (A,B,C);

If 0 < A < A_0, then there exist exactly three solutions (A,B,C).

Let f(B) = (sin A)^2 + (sin B)^2 - sin(A+B), then we must have f(B) = f'(B) = 0. Taking into account the conditions (sin A)^2 + (cos A)^2 = (sin B)^2 + (cos B)^2 = 1, we have a system of four quadratic equations in four variables sin A, cos A, sin B, cos B.

Links

Table of n, a(n) for n=0..89.

Jianing Song, Locus of (A,B) satisfying (sin A)^2 + (sin B)^2 = sin(A+B)

Index entries for transcendental numbers.

Formula

Let (A_0,B_0,C_0) be the nontrivial solution to (sin A_0)^2 + (sin B)^2 = sin C (i.e., C != Pi/2).

Then (sin A_0)^2 is the smallest real root to 128*x^5 - 176*x^4 - 200*x^3 + 91*x^2 + 98*x - 5 = 0. Equivalently, (cos A_0)^2 is the largest real root to 128*x^5 - 464*x^4 + 376*x^3 + 285*x^2 - 384*x + 64 = 0.

And (sin B_0)^2 is the unique real root to 256*x^5 - 400*x^4 + 312*x^3 - 131*x^2 + 30*x - 3 = 0. Equivalently, (cos B_0)^2 is the unique real root to 256*x^5 - 880*x^4 + 1272*x^3 - 965*x^2 + 384*x - 64 = 0. Note that cos B_0 is negative.

Example

A_0 = 0.22329653388455684579... radians, B_0 = 2.56825148044394043826... radians.

Mathematica

RealDigits[ArcSin[Root[128*x^10 - 176*x^8 - 200*x^6 + 91*x^4 + 98*x^2 - 5, 4]], 10, 120][[1]] (* Amiram Eldar, Apr 14 2026 *)

Prog

(PARI) my(s = solve(s=0, 0.5, 128*s^10 - 176*s^8 - 200*s^6 + 91*s^4 + 98*s^2 - 5)); asin(s)
(PARI) asin(polrootsreal(128*x^10 - 176*x^8 - 200*x^6 + 91*x^4 + 98*x^2 - 5)[4]) \\ Charles R Greathouse IV, May 18 2026

Crossrefs

Cf. A384807 (A_0 in degrees).

Sequence in context: A016012 A069138 A179592 * A058671 A016002 A098189

Adjacent sequences: A384802 A384803 A384804 * A384806 A384807 A384808

Keyword

nonn,cons,easy

Author

Jianing Song, Jun 10 2025

Status

approved