Answer
The missing quantities for the triangle are $ c=39.5,B={{54}^{{}^\circ }},C={{92}^{{}^\circ }}$ or $ c=13.7,B={{126}^{{}^\circ }},C={{20}^{{}^\circ }}$.
Work Step by Step
By using the sine rule,
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
For B,
$\begin{align}
& \sin B=\frac{b\times \sin A}{a} \\
& \sin B=\frac{32\times \sin {{34}^{{}^\circ }}}{22} \\
& \sin B=0.8134 \\
& B\approx {{54}^{{}^\circ }},{{126}^{{}^\circ }}
\end{align}$
Now, use the angle sum property to find the value of C.
$\begin{align}
& A+B+C={{180}^{{}^\circ }} \\
& {{34}^{{}^\circ }}+{{54}^{{}^\circ }}+C={{180}^{{}^\circ }} \\
& C={{92}^{{}^\circ }}
\end{align}$
Or
$\begin{align}
& {{34}^{{}^\circ }}+{{126}^{{}^\circ }}+C={{180}^{{}^\circ }} \\
& C={{180}^{{}^\circ }}-{{160}^{{}^\circ }} \\
& C={{20}^{{}^\circ }}
\end{align}$
Applying sine rule for $ C={{92}^{{}^\circ }}$
$\begin{align}
& c=22\frac{\sin {{92}^{{}^\circ }}}{\sin {{34}^{{}^\circ }}} \\
& =39.5
\end{align}$
Then, apply sine rule for $ C={{20}^{{}^\circ }}$
$\begin{align}
& c=22\frac{\sin {{20}^{{}^\circ }}}{\sin {{34}^{{}^\circ }}} \\
& =13.7
\end{align}$
Hence, the missing quantities for the triangle are $ c=39.5,B={{54}^{{}^\circ }},C={{92}^{{}^\circ }}$ or $ c=13.7,B={{126}^{{}^\circ }},C={{20}^{{}^\circ }}$.
