Given b=2,\ a=3, and B=40 (degrees), determine whether this information

pamangking8 2021-11-25 Answered
Given \(\displaystyle{b}={2},\ {a}={3}\), and \(\displaystyle{B}={40}\) (degrees), determine whether this information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangles.

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Expert Answer

oces3y
Answered 2021-11-26 Author has 5785 answers
Step 1
Given thet, the values are \(\displaystyle{b}={2},\ {a}={3}\) and \(\displaystyle{B}={40}^{{\circ}}\)
Recall that, the sine formula is \(\displaystyle{\frac{{{a}}}{{{\sin{{A}}}}}}={\frac{{{b}}}{{{\sin{{B}}}}}}={\frac{{{c}}}{{{\sin{{c}}}}}}\)
\(\displaystyle{\frac{{{\sin{{A}}}}}{{{a}}}}={\frac{{{\sin{{B}}}}}{{{b}}}}\)
\(\displaystyle{\sin{{A}}}={a}{\left({\frac{{{\sin{{B}}}}}{{{b}}}}\right)}\)
\(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{a}\times{\sin{{B}}}}}{{{b}}}}\right)}}\)
Step 2
Substitute \(\displaystyle{a}={3},\ {b}={2}\) and \(\displaystyle{B}={40}^{{\circ}}\) in \(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{a}\times{\sin{{B}}}}}{{{b}}}}\right)}}\)
\(\displaystyle{A}={{\sin}^{{-{1}}}{\left({\frac{{{3}\times{\sin{{\left({40}^{{\circ}}\right)}}}}}{{{2}}}}\right)}}\)
\(\displaystyle={{\sin}^{{-{1}}}{\left({0.9642}\right)}}\)
\(\displaystyle={1.3023}\times{\frac{{{180}^{{\circ}}}}{{\pi}}}\)
\(\displaystyle={74.6162}^{{\circ}}\)
\(\displaystyle{A}={74.6}^{{\circ}}\)
or
\(\displaystyle{A}={180}^{{\circ}}-{74.6}^{{\circ}}\)
\(\displaystyle{A}={105.4}^{{\circ}}\)
\(\displaystyle{\left(\because{105.4}^{{\circ}}+{40}^{{\circ}}{<}{180}^{{\circ}}\right)}\)</span>
Step 3
Let \(\displaystyle{A}={74.6}^{{\circ}}\) then the angle of C is,
\(\displaystyle{C}={180}^{{\circ}}-{74.6}^{{\circ}}-{40}^{{\circ}}\)
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 teo triangles.
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