Answer
$$a = 2.60{\text{yd}},{\text{ }}B = {45.1^ \circ },\,\,C = {93.5^ \circ }\,$$
Work Step by Step
$$\eqalign{
& A = {41.4^ \circ },\,\,\,b = 2.78{\text{yd,}}\,\,\,c = 3.92{\text{yd}} \cr
& {\text{Use the Law of cosines to find }}a \cr
& {a^2} = {b^2} + {c^2} - 2bc\cos A \cr
& {\text{Substitute}} \cr
& {a^2} = {\left( {2.78} \right)^2} + {\left( {3.92} \right)^2} - 2\left( {2.78} \right)\left( {3.92} \right)\cos \left( {{{41.4}^ \circ }} \right) \cr
& {\text{Use a calculator}} \cr
& {a^2} \approx 6.74597 \cr
& {\text{Take square roots and choose the positive root}} \cr
& a \approx 2.60{\text{yd}} \cr
& \cr
& {\text{Calculate the angle }}B{\text{ using the law of sines}} \cr
& \frac{b}{{\sin B}} = \frac{a}{{\sin A}} \cr
& \sin B = \frac{{b\sin A}}{a} \cr
& \sin B = \frac{{2.78\sin \left( {{{41.4}^ \circ }} \right)}}{{2.60}} \cr
& {\text{Use a calculator}} \cr
& \sin B \approx 0.70709 \cr
& B \approx {\sin ^{ - 1}}\left( {0.70709} \right) \cr
& B = {45.1^ \circ } \cr
& \cr
& {\text{Calculate }}C \cr
& C = {180^ \circ } - A - B \cr
& C = {180^ \circ } - {41.4^ \circ } - {45.1^ \circ } \cr
& C = {93.5^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& a = 2.60{\text{yd}},{\text{ }}B = {45.1^ \circ },\,\,C = {93.5^ \circ }\, \cr} $$
