Answer
The two possible set of values are $A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4\text{ or }A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4$.
Work Step by Step
For any triangle,
The law of sines states that:
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\operatorname{sinC}}$
The law of cosines states that:
$\begin{align}
& {{a}^{2}}={{b}^{2}}+{{c}^{2}}-2\cdot b\cdot c\cdot \cos A \\
& {{b}^{2}}={{a}^{2}}+{{c}^{2}}-2\cdot a\cdot c\cdot \cos B \\
& {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2\cdot a\cdot b\cdot \cos C \\
\end{align}$
Using the law of sines we get,
$\begin{align}
& \frac{a}{\sin A}=\frac{b}{\sin B} \\
& \frac{12.4}{\sin A}=\frac{8.7}{\sin 37{}^\circ } \\
& 8.7\left( \sin A \right)=12.4\left( \sin 37{}^\circ \right) \\
& \sin A=\frac{12.4\left( \sin 37{}^\circ \right)}{8.7}
\end{align}$
It can further be simplified to get a measure of angle C.
$\begin{align}
& \sin A=0.85 \\
& A={{\sin }^{-1}}\left( 0.85 \right) \\
& A\approx 59{}^\circ,121{}^\circ
\end{align}$
Now using the angle sum property for both values of A we will get the measure of angle C.
For $A=59{}^\circ $,
$\begin{align}
& 59{}^\circ +37{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -96{}^\circ \\
& C=84{}^\circ
\end{align}$
For $A=121{}^\circ $,
$\begin{align}
& 121{}^\circ +37{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -158{}^\circ \\
& C=22{}^\circ
\end{align}$
Now using the law of sines we will obtain two values of side c for different angles C:
For $C=84{}^\circ $
$\begin{align}
& \frac{8.7}{\sin 37{}^\circ }=\frac{c}{\sin 84{}^\circ } \\
& c\cdot \sin 37{}^\circ =8.7\left( \sin 84{}^\circ \right) \\
& c=\frac{8.7\left( \sin 84{}^\circ \right)}{\sin 37{}^\circ }
\end{align}$
This gives $c\approx 14.4$.
For $C=22{}^\circ $,
$\begin{align}
& \frac{8.7}{\sin 37{}^\circ }=\frac{c}{\sin 22{}^\circ } \\
& c\cdot \sin 37{}^\circ =8.7\left( \sin 22{}^\circ \right) \\
& c=\frac{8.7\left( \sin 22{}^\circ \right)}{\sin 37{}^\circ }
\end{align}$
This gives $c\approx 5.4$.
So,
$\begin{align}
& A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4 \\
& A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4 \\
\end{align}$
Hence, $A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4\text{ or }A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4$ are the two sets of possible values.
