Answer
The required values are $C=162{}^\circ,a\approx 33.8$, and $b\approx 67.3$.
Work Step by Step
At first we will find C.
Property of a triangle:
Sum of three angles is $A+B+C=180{}^\circ $
$\begin{align}
& A+B+C=180{}^\circ \\
& 6{}^\circ +12{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -18{}^\circ \\
& C=162{}^\circ
\end{align}$
Now, to find the remaining sides we will:
Use the ratio $\frac{c}{\sin C}$ or $\frac{100}{\sin 162{}^\circ }$,
Now we will use the law of sines to find a.
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{a}{\sin 6{}^\circ }=\frac{100}{\sin 162{}^\circ } \\
& a=\frac{100\sin 6{}^\circ }{\sin 162{}^\circ } \\
& a=33.8
\end{align}$
Again use the law of sines to find b
$\begin{align}
& \frac{b}{\sin B}=\frac{c}{\sin C} \\
& \frac{b}{\sin 12{}^\circ }=\frac{100}{\sin 162{}^\circ } \\
& b=\frac{100\sin 12{}^\circ }{\sin 162{}^\circ } \\
& b=67.3
\end{align}$
The solution is $C=162{}^\circ,a\approx 33.8$, and $b\approx 67.3$.
