Answer
$${A_1} = {72.16^ \circ },\,\,\,\,\,\,{A_2} = {107.84^ \circ },\,\,\,\,\,\,\,{C_1} = {59.64^ \circ },\,\,\,\,{C_2} = {23.96^ \circ }$$
Work Step by Step
$$\eqalign{
& B = {48.2^ \circ },\,\,\,a = 890{\text{cm,}}\,\,\,b = 697{\text{cm}} \cr
& \cr
& {\text{Use the law of sines to find the angle of }}A \cr
& \frac{a}{{\sin A}} = \frac{b}{{\sin B}} \cr
& \sin A = \frac{{a\sin B}}{b} \cr
& \sin A = \frac{{890\sin \left( {{{48.2}^ \circ }} \right)}}{{697}} \cr
& {\text{Use a calculator}} \cr
& \sin A = 0.951899 \cr
& \cr
& {\text{There are two }}\,{\text{angles }}B\,{\text{between }}\,{{\text{0}}^ \circ }{\text{ and 18}}{{\text{0}}^ \circ }{\text{ that satisfy this }} \cr
& {\text{condition}}{\text{.}} \cr
& {A_1} = {\sin ^{ - 1}}\left( {0.951899} \right) \cr
& {\text{Use the inverse sine function}} \cr
& {A_1} = {72.16^ \circ } \cr
& \cr
& {\text{Supplementary angles have the same sine value}},{\text{ so another }} \cr
& {\text{possible value of angle }}A{\text{ is}} \cr
& {A_2} = {180^ \circ } - {72.16^ \circ } \cr
& {A_2} = {107.84^ \circ } \cr
& \cr
& {\text{Calculating }}{C_1} \cr
& {C_1} = {180^ \circ } - B - {A_1} \cr
& {C_1} = {180^ \circ } - {48.2^ \circ } - {72.16^ \circ } \cr
& {C_1} = {59.64^ \circ } \cr
& \cr
& {\text{Calculating }}{C_2} \cr
& {C_2} = {180^ \circ } - B - {A_2} \cr
& {C_2} = {180^ \circ } - {48.2^ \circ } - {107.84^ \circ } \cr
& {C_2} = {23.96^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& {A_1} = {72.16^ \circ },\,\,\,\,\,\,{A_2} = {107.84^ \circ },\,\,\,\,\,\,\,{C_1} = {59.64^ \circ },\,\,\,\,{C_2} = {23.96^ \circ } \cr} $$