Answer
$$B = 74.6^\circ ,\,\,\,C = 43.7^\circ ,\,\,\,\,c = 62{\text{m}}$$
Work Step by Step
$$\eqalign{
& A = {\text{61}}.{\text{7}}^\circ ,a = {\text{78}}.{\text{9 m}},b = {\text{86}}.{\text{4 m}} \cr
& {\text{Calculate the angle }}B{\text{ using the law of sines}} \cr
& \frac{{\sin B}}{b} = \frac{{\sin A}}{a} \cr
& \sin B = \frac{{b\sin A}}{a} \cr
& \sin B = \frac{{{\text{86}}.{\text{4}}\sin \left( {{\text{61}}.{\text{7}}^\circ } \right)}}{{{\text{78}}.{\text{9}}}} \cr
& {\text{Use a calculator}} \cr
& \sin B \approx 0.9641729194 \cr
& {\text{Use the inverse sine function}} \cr
& B \approx 74.6^\circ \cr
& \cr
& {\text{Calculate the angle }}C \cr
& C = 180^\circ - A - B \cr
& C = 180^\circ - {\text{61}}.{\text{7}}^\circ - 74.6^\circ \cr
& C = 43.7^\circ \cr
& \cr
& {\text{Calculate the side }}c{\text{ using the law of sines}} \cr
& \frac{c}{{\sin C}} = \frac{a}{{\sin A}} \cr
& c = \frac{{a\sin C}}{{\sin A}} \cr
& c = \frac{{{\text{78}}.{\text{9}}\sin \left( {43.7^\circ } \right)}}{{\sin \left( {{\text{61}}.{\text{7}}^\circ } \right)}} \cr
& c \approx 62{\text{m}} \cr
& \cr
& {\text{Answer}} \cr
& B = 74.6^\circ ,\,\,\,C = 43.7^\circ ,\,\,\,\,c = 62{\text{m}} \cr} $$
