Two sides and an angle are given below. Determine whether the given information

Two sides and an angle are given below. Determine whether the given information results in one triangles, or no triangle at all. Solve any resulting triangle(s).
$$\displaystyle{b}={4},\ {c}={5},\ {B}={30}^{{\circ}}$$
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A) A single triangle is produced, where $$\displaystyle{C}\approx?^{{\circ}},\ {A}\approx?^{{\circ}}$$, and $$\displaystyle{a}\approx?$$
B) Two triangles are produced, where the triangle with the smaller angle C has $$\displaystyle{C}_{{{1}}}\approx?^{{\circ}},\ {A}_{{{1}}}\approx?^{{\circ}}$$, and $$\displaystyle{a}_{{{1}}}\approx?$$, and the triangle with the larger angle C has $$\displaystyle{C}_{{{2}}}\approx?^{{\circ}},\ {A}_{{{2}}}\approx?^{{\circ}},$$ and $$\displaystyle{a}_{{{2}}}\approx?$$
C) No triangles are produced.

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William Yazzie
Step 1
This problem can be solved using Law of sines of triangle. The Law of Sines is the relationship between the sides and angles of non-right triangles.
Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.

for the triangle $$\displaystyle\triangle{A}{B}{C}$$ shown in figure with side lengths a, b, c the law of sines is given as,
$$\displaystyle{\frac{{{a}}}{{{\sin{{A}}}}}}={\frac{{{b}}}{{{\sin{{B}}}}}}={\frac{{{c}}}{{{\sin{{C}}}}}}$$
Step 2
In the question, side $$\displaystyle{b}={4},\ {c}={5},\ {B}={30}^{{\circ}}$$
applying law of sines,
$$\displaystyle{\frac{{{4}}}{{{\sin{{30}}}^{{\circ}}}}}={\frac{{{5}}}{{{\sin{{C}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{4}}}{{{0.5}}}}={\frac{{{5}}}{{{\sin{{C}}}}}}$$
$$\displaystyle\Rightarrow{8}={\frac{{{5}}}{{{\sin{{C}}}}}}$$
$$\displaystyle\Rightarrow{\sin{{C}}}={\frac{{{5}}}{{{8}}}}={0.625}$$
$$\displaystyle\Rightarrow{C}={{\sin}^{{-{1}}}{\left({0.625}\right)}}={38.68}^{{\circ}}$$
The sum of angles of triangle $$\displaystyle={180}^{{\circ}}$$
therefore
$$\displaystyle\angle{A}={180}-{38.68}-{30}$$
$$\displaystyle\angle{A}={11.32}^{{\circ}}$$
now using Law of sines we can find length of third side.
$$\displaystyle{\frac{{{a}}}{{{\sin{{A}}}}}}={\frac{{{b}}}{{{\sin{{B}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{a}}}{{{\sin{{111.32}}}}}}={\frac{{{4}}}{{{\sin{{30}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{a}}}{{{0.931}}}}={\frac{{{4}}}{{{0.5}}}}$$
$$\displaystyle\Rightarrow{a}={\frac{{{4}}}{{{0.5}}}}\times{0.931}={7.45}$$
Step 3
Now let us see if we can have another triangle. We have $$\displaystyle{B}={30}^{{\circ}}$$ given
Now subtract already calculated C from $$\displaystyle{180}^{{\circ}}$$
$$\displaystyle\Rightarrow{C}_{{{2}}}={180}^{{\circ}}-{30}^{{\circ}}-{141.32}^{{\circ}}$$
therefore the third angle will be,
$$\displaystyle{A}_{{{2}}}={180}^{{\circ}}-{30}^{{\circ}}-{141.32}^{{\circ}}$$
$$\displaystyle{A}_{{{2}}}={8.68}^{{\circ}}$$
$$\displaystyle{\frac{{{a}_{{{2}}}}}{{{\sin{{A}}}}}}={\frac{{{b}}}{{{\sin{{B}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{a}_{{{2}}}}}{{{\sin{{8.68}}}}}}={\frac{{{4}}}{{{\sin{{30}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{a}_{{{2}}}}}{{{0.15}}}}={\frac{{{4}}}{{{0.5}}}}$$
$$\displaystyle\Rightarrow{a}_{{{2}}}={\frac{{{4}}}{{{0.5}}}}\times{0.15}={1.207}$$
Step 4
Therefore there are two triangles possible for the given measurements $$\displaystyle{b}={4},\ {c}={5},\ {B}={30}^{{\circ}}$$
Triangle 1
$$\displaystyle{C}_{{{1}}}={38.68}^{{\circ}}$$
$$\displaystyle{A}_{{{1}}}={111.32}^{{\circ}}$$
$$\displaystyle{a}_{{{1}}}={7.45}$$
Triangle 2
$$\displaystyle{C}_{{{2}}}={141.32}^{{\circ}}$$
$$\displaystyle{A}_{{{2}}}={8.68}^{{\circ}}$$
$$\displaystyle{a}_{{{2}}}={1.207}$$