[tex]\displaystyle \\
Se da: \\
\Delta ABC ~cu~Aria = 12 \sqrt{6} ~cm^2\\
AB = 16~cm~~si ~~AC=3\sqrt{6} ~cm \\ \\
Se~cere: \\
\ \textless \ A = ~? \\ \\
Rezolvare: \\ \\
Aria = \frac{AB\times AC \times \sin A }{2} \\ \\
\frac{16 \times 3 \sqrt{6} \times \sin A }{2} = 12 \sqrt{6} \\ \\
8 \times 3 \sqrt{6} \times \sin A = 12 \sqrt{6} \\
24 \sqrt{6} \times \sin A = 12 \sqrt{6} \\ \\
\sin A = \frac{12 \sqrt{6}}{24 \sqrt{6} } =\frac{12 }{24 }=\boxed{\frac{1 }{2} }
[/tex]
[tex]\texttt{Unghiul A poate fi ascutit sau obtuz. } \\
\texttt{Rezulta ca unghiul A poate fi in cadranul 1 sau 2. } \\ \\
Solutia 1: \\
\ \textless \ A \in cadranului ~ 1 \\
\ \textless \ A = \arcsin ( \frac{1}{2} ) = \boxed{30^o } \\ \\
Solutia 2: \\
\ \textless \ A \in cadranului ~ 2\\
\ \textless \ A = \arcsin ( \frac{1}{2} ) = \boxed{150^o } \\ \\
Verificare: \displaystyle \\ \\
\sin 150^o = \sin (180^o-150^o) = \sin 30^o = \frac{1}{2}[/tex]
