The law of sines in hyperbolic geometry What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the constant only in Euclidean and spherical geometry.

As given by Will Jagy, k must be inside the argument:
$$ \frac{\sin A}{\sinh(a/k)} =  \frac{\sin B}{\sinh(b/k)} =  \frac{\sin C}{\sinh(c/k)} $$
This is the Law of Hyperbolic trigonometry where k is the pseudoradius, constant Gauss curvature $K= -1/k^2$. Please also refer to " Pan-geometry", a set of relations mirrored from spherical to hyerbolic, typified by (sin,cos) -> (sinh,cosh).. in Roberto Bonola's book on Non-euclidean Geometry.
There is nothing imaginary about pseudoradius.It is as real,palpable and solid as the radius of sphere in spherical trigonometry, after hyperbolic geometry has been so firmly established.
I wish practice of using $ K=-1$ should be done away with,always using $ K = -1/k^2 $ or $ K = -1/a^2 $instead. 
