stylistic:tone
The case of surfaces of constant curvature

Define a generalized sine function, depending also on a real parameter κ {\displaystyle \kappa }: sin κ (x) = x − κ 3 ! x 3 + κ 2 5 ! x 5 − κ 3 7 ! x 7 + ⋯ = ∑ n = 0 ∞ (− 1) n κ n (2 n + 1) ! x 2 n + 1. {\displaystyle \sin _{\kappa }(x)=x-{\frac {\kappa }{3!}}x^{3}+{\frac {\kappa ^{2}}{5!}}x^{5}-{\frac {\kappa ^{3}}{7!}}x^{7}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}\kappa ^{n}}{(2n+1)!}}x^{2n+1}.}

The law of sines in constant curvature κ {\displaystyle \kappa } reads as sin A sin κ a = sin B sin κ b = sin C sin κ c. {\displaystyle {\frac {\sin A}{\sin _{\kappa }a}}={\frac {\sin B}{\sin _{\kappa }b}}={\frac {\sin C}{\sin _{\kappa }c}}\,.}

By substituting κ = 0 {\displaystyle \kappa =0}, κ = 1 {\displaystyle \kappa =1}, and κ = − 1 {\displaystyle \kappa =-1}, one obtains respectively sin 0 (x) = x {\displaystyle \sin _{0}(x)=x}, sin 1 (x) = sin x {\displaystyle \sin _{1}(x)=\sin x}, and sin − 1 (x) = sinh x {\displaystyle \sin _{-1}(x)=\sinh x}, that is, the Euclidean, spherical, and hyperbolic cases of the law of sines described above.

Let p κ (r) {\displaystyle p_{\kappa }(r)} indicate the circumference of a circle of radius r {\displaystyle r} in a space of constant curvature κ {\displaystyle \kappa }. Then p κ (r) = 2 π sin κ (r) {\displaystyle p_{\kappa }(r)=2\pi \sin _{\kappa }(r)}. Therefore, the law of sines can also be expressed as: sin A p κ (a) = sin B p κ (b) = sin C p κ (c). {\displaystyle {\frac {\sin A}{p_{\kappa }(a)}}={\frac {\sin B}{p_{\kappa }(b)}}={\frac {\sin C}{p_{\kappa }(c)}}\,.}

This formulation was discovered by János Bolyai.