Use the Law of Cosines to solve the triangles. Round lengths to the nearest tent

mronjo7n 2021-11-19 Answered
Use the Law of Cosines to solve the triangles. Round lengths to the nearest tenth and angle measures to the nearest degree.
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Expert Answer

romancemrf
Answered 2021-11-20 Author has 558 answers
Step 1
Given that,
\(\displaystyle{A}={22}^{{\circ}},\ {b}={6},\ {c}={15}\)
using Cosine rule, \(\displaystyle{\cos{{A}}}={\frac{{{b}^{{{2}}}+{c}^{{{2}}}-{a}^{{{2}}}}}{{{2}{b}{c}}}}\)
\(\displaystyle\Rightarrow{\cos{{\left({22}\right)}}}={\frac{{{\left({6}\right)}^{{{2}}}+{\left({15}\right)}^{{{2}}}-{a}^{{{2}}}}}{{{2}{\left({6}\right)}{\left({15}\right)}}}}\)
\(\displaystyle\Rightarrow{0.9272}={\frac{{{36}+{225}-{a}^{{{2}}}}}{{{180}}}}\)
\(\displaystyle\Rightarrow{166.893}={261}-{a}^{{{2}}}\)
\(\displaystyle\Rightarrow{a}^{{{2}}}={94.1069}\)
\(\displaystyle\Rightarrow{a}={9.7009}\)
\(\displaystyle\Rightarrow{a}\approx{10}\)
Step 2
Using Sine rule, \(\displaystyle{\frac{{{\sin{{A}}}}}{{{a}}}}={\frac{{{\sin{{B}}}}}{{{b}}}}\)
\(\displaystyle\Rightarrow{\frac{{{\sin{{\left({22}\right)}}}}}{{{10}}}}={\frac{{{\sin{{B}}}}}{{{6}}}}\)
\(\displaystyle\Rightarrow{\sin{{B}}}={\frac{{{6}{\sin{{\left({22}\right)}}}}}{{{10}}}}\)
\(\displaystyle\Rightarrow{B}\approx{13}^{{\circ}}\)
using angle sum property of a triangle,
\(\displaystyle{C}={180}-{\left({A}+{B}\right)}\)
\(\displaystyle={180}-{\left({22}+{13}\right)}\)
\(\displaystyle={145}^{{\circ}}\)
Therefore, \(\displaystyle{a}={10},\ {B}={13}^{{\circ}},\ {C}={145}^{{\circ}}\)
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