Answer
$$\,c = 4.28{\text{mi, }}A = {48.8^ \circ },\,\,B = {71.5^ \circ }$$
Work Step by Step
$$\eqalign{
& C = {\text{59}}.{\text{7}}^\circ ,\,\,\,a = {\text{3}}.{\text{73mi}},\,\,b = {\text{4}}.{\text{7}}0{\text{mi}} \cr
& {\text{Use the Law of cosines to find }}c \cr
& {c^2} = {a^2} + {b^2} - 2ab\cos C \cr
& {\text{Substitute}} \cr
& {c^2} = {\left( {{\text{3}}.{\text{73}}} \right)^2} + {\left( {{\text{4}}.{\text{7}}0} \right)^2} - 2\left( {{\text{3}}.{\text{73}}} \right)\left( {{\text{4}}.{\text{7}}0} \right)\cos \left( {{\text{59}}.{\text{7}}^\circ } \right) \cr
& {\text{Use a calculator}} \cr
& {c^2} \approx 18.31315245 \cr
& {\text{Take square roots and choose the positive root}} \cr
& c \approx 4.28{\text{mi}} \cr
& \cr
& {\text{Calculate the angle }}A{\text{ using the law of sines}} \cr
& \frac{a}{{\sin A}} = \frac{c}{{\sin C}} \cr
& \sin A = \frac{{a\sin C}}{c} \cr
& \sin A = \frac{{{\text{3}}.{\text{73}}\sin \left( {{\text{59}}.{\text{7}}^\circ } \right)}}{{4.28}} \cr
& {\text{Use a calculator}} \cr
& \sin A \approx 0.7524451878 \cr
& {\text{Use the inverse sine function}} \cr
& A = {48.8^ \circ } \cr
& \cr
& {\text{Calculate }}B \cr
& B = {180^ \circ } - A - C \cr
& B = {180^ \circ } - {48.8^ \circ } - {\text{59}}.{\text{7}}^\circ \cr
& B = {71.5^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& \,c = 4.28{\text{mi, }}A = {48.8^ \circ },\,\,B = {71.5^ \circ } \cr} $$
