Answer
The required values are $A=82{}^\circ,B=41{}^\circ,C=57{}^\circ,c\approx 255.7$.
Work Step by Step
Using the law of sines we will obtain:
$\begin{align}
& \frac{a}{\sin A}=\frac{b}{\sin B} \\
& \frac{300}{\sin 2\theta }=\frac{200}{\sin \theta } \\
& 200\sin 2\theta =300\sin \theta
\end{align}$
Using the identity: $\sin 2\theta =2\sin \theta \cos \theta $
$400\sin \theta \cos \theta =300\sin \theta $
Now,
$\begin{align}
& \cos \theta =\frac{300\sin \theta }{400\sin \theta } \\
& \cos \theta =\frac{3}{4} \\
& \theta \approx 41{}^\circ \\
& 2\theta \approx 82{}^\circ
\end{align}$
Now, we will find C, by using the property that the sum of all angles is $180{}^\circ $
So,
$\begin{align}
& A+B+C=180{}^\circ \\
& 82{}^\circ +41{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -123{}^\circ \\
& C=57{}^\circ
\end{align}$
Using the law of sines, we will find c.
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{300}{\sin 82{}^\circ }=\frac{c}{\sin 57{}^\circ } \\
& c=\frac{300\sin 57{}^\circ }{\sin 82{}^\circ } \\
& c\approx 255.7
\end{align}$
Hence, the required values are $A=82{}^\circ,B=41{}^\circ,C=57{}^\circ,c\approx 255.7$.
