Hyperbolic side-side-angle ambiguity ($c_{\pm }=\ln (3\pm \sqrt{2})$)

Two corresponding side lengths and an equal nonincluded angle need not determine a hyperbolic triangle. Set $A=\pi /6$, $\cosh a=3/2$, $\cosh b=2$. The hyperbolic law of cosines has two distinct positive solutions $c=\ln (3\pm \sqrt{2})$. The two triangles share $a,b,A$ but have different third sides. An included angle instead determines the third side by the cosine rule.