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

hrostentsp6 2021-11-17 Answered
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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Expert Answer

William Yazzie
Answered 2021-11-18 Author has 291 answers
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.
image
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
\(\displaystyle{C}_{{{2}}}={141.32}^{{\circ}}\)
\(\displaystyle{A}_{{{2}}}={8.68}^{{\circ}}\)
\(\displaystyle{a}_{{{2}}}={1.207}\)
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