[tex]\displaystyle\\
\text{Folosim teorema sinusurilor: }~~~ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \\\\
\text{Aplicam formula in triunghiul ABC.}\\\\
\frac{AC}{\sin B}= \frac{AB}{\sin C}\\\\
\Longrightarrow \frac{AB}{AC}= \frac{\sin C}{\sin B}=\frac{\sin \dfrac{\pi}{4} }{\sin \dfrac{\pi}{6} }=\frac{\sin 45^o}{\sin 30^o}= \frac{ \dfrac{ \sqrt{2} }{2} }{ \dfrac{1}{2}}= \frac{ \sqrt{2} }{2}\times \frac{2}{1} =\frac{ 2\sqrt{2} }{2}=\boxed{\bf \sqrt{2}}[/tex]
