Answer
$$B = {67^ \circ }45',\,\,\,\,\,\,a = 22.03{\text{mm}},\,\,\,\,\,\,\,b = 37.5{\text{mm}}$$
Work Step by Step
$$\eqalign{
& C = {79^ \circ }18',\,\,\,c = 39.81{\text{mm,}}\,\,\,A = {32^ \circ }57' \cr
& \cr
& {\text{Find }}B \cr
& B = {180^ \circ } - A - C \cr
& B = {180^ \circ } - {32^ \circ }57' - {79^ \circ }18' \cr
& B = {67^ \circ }45' \cr
& \cr
& {\text{Use the law of sines to find side }}a \cr
& \frac{a}{{\sin A}} = \frac{c}{{\sin C}} \cr
& a = \frac{{c\sin A}}{{\sin C}} \cr
& a = \frac{{39.81\sin \left( {{{32}^ \circ }57'} \right)}}{{\sin \left( {{{79}^ \circ }18'} \right)}} \cr
& {\text{Use a calculator}} \cr
& a = 22.03{\text{mm}} \cr
& \cr
& {\text{Use the law of sines to find side }}b \cr
& \frac{b}{{\sin B}} = \frac{c}{{\sin C}} \cr
& b = \frac{{c\sin B}}{{\sin C}} \cr
& b = \frac{{39.81\sin \left( {{{67}^ \circ }45'} \right)}}{{\sin \left( {{{79}^ \circ }18'} \right)}} \cr
& {\text{Use a calculator}} \cr
& b = 37.5{\text{mm}} \cr
& \cr
& {\text{Answer}} \cr
& B = {67^ \circ }45',\,\,\,\,\,\,a = 22.03{\text{mm}},\,\,\,\,\,\,\,b = 37.5{\text{mm}} \cr} $$
