Answer
$$A = 55^\circ 17',\,\,\,B = 94^\circ 53',\,\,\,b = 10.43{\text{m}}$$
Work Step by Step
$$\eqalign{
& C = {\text{29}}^\circ {\text{5}}0\prime ,\,a = {\text{8}}.{\text{61 m}},\,\,c = {\text{5}}.{\text{21 m}} \cr
& {\text{Use the law of sines to find the angle of }}A \cr
& \frac{{\sin A}}{a} = \frac{{\sin C}}{c} \cr
& \sin A = \frac{{a\sin C}}{c} \cr
& {\text{Substituting }} \cr
& \sin A = \frac{{\left( {{\text{8}}.{\text{61}}} \right)\sin \left( {{\text{29}}^\circ {\text{5}}0\prime } \right)}}{{{\text{5}}.{\text{21}}}} \cr
& {\text{Use a calculator}} \cr
& \sin A \approx 0.8221289443 \cr
& A \approx {\sin ^{ - 1}}\left( {0.8221289443} \right) \cr
& A \approx 55^\circ 17' \cr
& \cr
& {\text{Calculate the angle of }}B \cr
& B = 180^\circ - A - C \cr
& B = 180^\circ - 55^\circ 17' - {\text{29}}^\circ {\text{5}}0\prime \cr
& B = 94^\circ 53' \cr
& \cr
& {\text{Use the law of sines to find side }}b \cr
& \frac{b}{{\sin B}} = \frac{a}{{\sin A}} \cr
& b = \frac{{a\sin B}}{{\sin A}} \cr
& b = \frac{{{\text{8}}.{\text{61}}\sin \left( {94^\circ 53'} \right)}}{{\sin \left( {55^\circ 17'} \right)}} \cr
& b \approx 10.43{\text{m}} \cr
& \cr
& {\text{Answer}} \cr
& A = 55^\circ 17',\,\,\,B = 94^\circ 53',\,\,\,b = 10.43{\text{m}} \cr} $$
