Answer
The required values are $A=96{}^\circ,B=48{}^\circ,C=36{}^\circ,c\approx 237.3$.
Work Step by Step
Using the law of sines, we will obtain:
$\begin{align}
& \frac{a}{\sin A}=\frac{b}{\sin B} \\
& \frac{400}{\sin 2\theta }=\frac{300}{\sin \theta } \\
& 300\sin 2\theta =400\sin \theta
\end{align}$
Using the identity: $\sin 2\theta =2\sin \theta \cos \theta $, we get
So, $600\sin \theta \cos \theta =400\sin \theta $
Now,
$\begin{align}
& \cos \theta =\frac{400\sin \theta }{600\sin \theta } \\
& \cos \theta =\frac{2}{3} \\
& \theta =48{}^\circ \\
& 2\theta =96{}^\circ
\end{align}$
Now, to find C, we will use the rule that the sum of all angles is $180{}^\circ $
So,
$\begin{align}
& A+B+C=180{}^\circ \\
& 96{}^\circ +48{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -144{}^\circ \\
& C=36{}^\circ
\end{align}$
Using the law of sines, we will find c.
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{400}{\sin 96{}^\circ }=\frac{c}{\sin 36{}^\circ } \\
& c=\frac{400\sin 36{}^\circ }{\sin 96{}^\circ } \\
& c\approx 237.3
\end{align}$
Hence, the required values are $A=96{}^\circ,B=48{}^\circ,C=36{}^\circ,c\approx 237.3$.
