In order to calculate the length of the third side, let's use the law of cosines:
[tex]\begin{gathered} b^2=a^2+c^2-2\cdot a\cdot c\cdot\cos (B) \\ b^2=8.32^2+5.97^2-2\cdot8.32\cdot5.97\cdot\cos (104.7\degree) \\ b^2=69.2224+35.6409-99.3408\cdot(-0.253758) \\ b^2=130.0718 \\ b=11.405 \end{gathered}[/tex]
Now, let's calculate angle A using law of sines:
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B} \\ \frac{8.32}{\sin A}=\frac{11.405}{0.9672677} \\ \sin A=\frac{0.9672677\cdot8.32}{11.405}=0.70562624 \\ A=\sin ^{-1}(0.70562624) \\ A=44.88\degree \end{gathered}[/tex]
Since the sum of internal angles in any triangle is equal to 180°, we have:
[tex]\begin{gathered} A+B+C=180 \\ 44.88+104.7+C=180 \\ 149.58+C=180 \\ C=180-149.58 \\ C=30.42\degree \end{gathered}[/tex]
Therefore the answer is option A:
A = 44.9°, C = 30.4°, b = 11.41 m
