Answer
$$A = {82^ \circ },\,\,\,\,B = {37^ \circ },\,\,\,\,C = {61^ \circ }$$
Work Step by Step
$$\eqalign{
& a = 9.3{\text{cm,}}\,\,\,b = 5.7{\text{cm,}}\,\,\,c = 8.2{\text{cm}} \cr
& \cr
& {\text{Use the law of cosines to solve for any angle of the triangle}} \cr
& {c^2} = {a^2} + {b^2} - 2ab\cos C \cr
& {\text{Solve for cos }}C \cr
& \cos C = \frac{{{a^2} + {b^2} - {c^2}}}{{2ab}} \cr
& {\text{Substitute}} \cr
& \cos C = \frac{{{{\left( {9.3} \right)}^2} + {{\left( {5.7} \right)}^2} - {{\left( {8.2} \right)}^2}}}{{2\left( {9.3} \right)\left( {5.7} \right)}} \cr
& {\text{Use a calculator}} \cr
& \cos C \approx 0.4880211281 \cr
& {\text{Use the inverse cosine function}} \cr
& C \approx {61^ \circ } \cr
& \cr
& {\text{Use the Law of sines to find the angle of }}A \cr
& \frac{{\sin A}}{a} = \frac{{\sin C}}{c} \cr
& \sin A = \frac{{a\sin C}}{c} \cr
& \sin A = \frac{{9.3\sin \left( {{{61}^ \circ }} \right)}}{{8.2}} \cr
& {\text{Use a calculator}} \cr
& \sin A \approx 0.991946441 \cr
& {\text{Use the inverse sine function}} \cr
& A = {82^ \circ } \cr
& \cr
& {\text{Calculate }}B \cr
& B = {180^ \circ } - A - C \cr
& B = {180^ \circ } - {82^ \circ } - {61^ \circ } \cr
& B = {37^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& A = {82^ \circ },\,\,\,\,B = {37^ \circ },\,\,\,\,C = {61^ \circ } \cr} $$
