Step 1

Given that, the values are \(\displaystyle{b}={2},\ {a}={3}\) and \(\displaystyle{B}={40}^{{\circ}}\)

Recall that, the sine formula is \(\displaystyle{\frac{{{a}}}{{{\sin{{A}}}}}}={\frac{{{b}}}{{{\sin{{B}}}}}}={\frac{{{c}}}{{{\sin{{c}}}}}}\)

\(\displaystyle{\frac{{{\sin{{A}}}}}{{{a}}}}={\frac{{{\sin{{B}}}}}{{{b}}}}\)

\(\displaystyle{\sin{{A}}}={a}{\left({\frac{{{\sin{{B}}}}}{{{b}}}}\right)}\)

\(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{a}\cdot{\sin{{B}}}}}{{{b}}}}\right)}}\)

Step 2

Substitute \(\displaystyle{a}={3},\ {b}={2}\) and \(\displaystyle{B}={40}^{{\circ}}\) in \(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{a}\cdot{\sin{{B}}}}}{{{b}}}}\right)}}\)

\(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{3}\cdot{\sin{{\left({40}^{{\circ}}\right)}}}}}{{{2}}}}\right)}}\)

\(\displaystyle={{\sin}^{{-{1}}}{\left({0.9642}\right)}}\)

\(\displaystyle={1.3023}\times{\frac{{{180}^{{\circ}}}}{{\pi}}}\)

\(\displaystyle={74.6162}^{{\circ}}\)

\(\displaystyle{A}={74.6}^{{\circ}}\)

(or)

\(\displaystyle{A}={180}^{{\circ}}-{74.6}^{{\circ}}\)

\(\displaystyle{A}={105.4}^{{\circ}}\)

\(\displaystyle{\left(\because{105.4}^{{\circ}}+{40}^{{\circ}}{<}{180}^{{\circ}}\right)}\)

Step 3

Let \(\displaystyle{A}={74.6}^{{\circ}}\) then the angle of C is,

\(\displaystyle{C}={180}^{{\circ}}-{74.6}^{{\circ}}-{40}^{{\circ}}\)

\(\displaystyle={65.4}^{{\circ}}\)

Use the sine formula to find the value of c.

\(\displaystyle{\frac{{{\sin{{B}}}}}{{{b}}}}={\frac{{{\sin{{C}}}}}{{{c}}}}\)

\(\displaystyle{c}={\frac{{{b}{\sin{{C}}}}}{{{\sin{{A}}}}}}\)

\(\displaystyle{c}={\frac{{{2}{\sin{{65.4}}}^{{\circ}}}}{{{\sin{{40}}}^{{\circ}}}}}\)

\(\displaystyle{c}={2.83}\)

Thus, the possible values of the triangle is \(\displaystyle{a}={3},\ {b}={2},\ {c}={2.83}\) and \(\displaystyle{A}={74.6}^{{\circ}},\ {B}={40}^{{\circ}},\ {C}={65.4}^{{\circ}}\)

Step 4

Let \(\displaystyle{A}={105.4}^{{\circ}}\) then the angle of C is,

\(\displaystyle{C}={180}^{{\circ}}-{105.4}^{{\circ}}-{40}^{{\circ}}\)

\(\displaystyle={34.6}^{{\circ}}\)

Use the sine formula to find the value of c.

\(\displaystyle{\frac{{{\sin{{B}}}}}{{{b}}}}={\frac{{{\sin{{C}}}}}{{{c}}}}\)

\(\displaystyle{c}={\frac{{{b}{\sin{{C}}}}}{{{\sin{{A}}}}}}\)

\(\displaystyle{c}={\frac{{{2}{\sin{{34.6}}}^{{\circ}}}}{{{\sin{{40}}}^{{\circ}}}}}\)

\(\displaystyle{c}={1.77}\)

Thus, the possible values of the triangle is

\(\displaystyle{a}={3},\ {b}={2},\ {c}={1.77}\) and \(\displaystyle{A}={105.4}^{{\circ}},\ {B}={40}^{{\circ}},\ {C}={34.6}^{{\circ}}\)

Therefore, the given values are possible for two triangles.