Answer
The required values are $B=42{}^\circ,a\approx 8.1,$ and $b\approx 8.1$.
Work Step by Step
At first we will find B.
Properties of a triangle:
Sum of three angles is $A+B+C=180{}^\circ $
$\begin{align}
& A+B+C=180{}^\circ \\
& 42{}^\circ +B+96{}^\circ =180{}^\circ \\
& B=180{}^\circ -138{}^\circ \\
& B=42{}^\circ
\end{align}$
Now, to find the remaining sides, we will
use the ratio $\frac{c}{\sin C}$ or $\frac{12}{\sin 96{}^\circ }$.
Now, we will use the law of sines to find a.
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{a}{\sin 42{}^\circ }=\frac{12}{\sin 96{}^\circ } \\
& a=\frac{12\sin 42{}^\circ }{\sin 96{}^\circ } \\
& a=8.1
\end{align}$
Again, use the law of sines to find b $\begin{align}
& \frac{b}{\sin B}=\frac{c}{\sin C} \\
& \frac{b}{\sin 42{}^\circ }=\frac{12}{\sin 96{}^\circ } \\
& b=\frac{12\sin 42{}^\circ }{\sin 96{}^\circ } \\
& b=8.1
\end{align}$
The solution is $B=42{}^\circ,a\approx 8.1,$ and $b\approx 8.1$.
