Step 1

The law of cosines is given by

\(\displaystyle{\cos{{A}}}={\frac{{{b}^{{{2}}}+{c}^{{{2}}}-{a}^{{{2}}}}}{{{2}{b}{c}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{a}^{{{2}}}+{c}^{{{2}}}-{b}^{{{2}}}}}{{{2}{a}{c}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{a}^{{{2}}}+{b}^{{{2}}}-{c}^{{{2}}}}}{{{2}{a}{b}}}}\)

Step 2

Given

\(\displaystyle{a}={5},\ {b}={5},\ {c}={5}\)

Using law of cosines,

\(\displaystyle{\cos{{A}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{25}+{25}-{25}}}{{{50}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle{A}={{\cos}^{{-{1}}}{\left({\frac{{{1}}}{{{2}}}}\right)}}\)

\(\displaystyle\Rightarrow{A}={60}^{{\circ}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{B}={60}^{{\circ}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{C}={60}^{{\circ}}\)

Therefore,

\(\displaystyle{A}={60}^{{\circ}}\)

\(\displaystyle{B}={60}^{{\circ}}\)

\(\displaystyle{C}={60}^{{\circ}}\)

Step 3

Given,

\(\displaystyle{a}={66},\ {b}={25},\ {c}={45}\)

Using law of cosines,

\(\displaystyle{\cos{{A}}}={\frac{{{\left({25}\right)}^{{{2}}}+{\left({45}\right)}^{{{2}}}-{\left({66}\right)}^{{{2}}}}}{{{2}{\left({25}\right)}{\left({45}\right)}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{625}+{2025}-{4356}}}{{{2250}}}}\)

\(\displaystyle{\cos{{A}}}=-{\frac{{{853}}}{{{1125}}}}\)

\(\displaystyle{A}={{\cos}^{{-{1}}}{\left(-{\frac{{{853}}}{{{1125}}}}\right)}}\)

\(\displaystyle\Rightarrow{A}\approx{139}^{{\circ}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{\left({66}\right)}^{{{2}}}+{\left({45}\right)}^{{{2}}}-{\left({25}\right)}^{{{2}}}}}{{{2}{\left({66}\right)}{\left({45}\right)}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{1439}}}{{{1485}}}}\)

\(\displaystyle\Rightarrow{B}\approx{14}^{{\circ}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{\left({66}\right)}^{{{2}}}+{\left({25}\right)}^{{{2}}}-{\left({45}\right)}^{{{2}}}}}{{{2}{\left({66}\right)}{\left({25}\right)}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{739}}}{{{825}}}}\)

\(\displaystyle\Rightarrow{C}\approx{26}^{{\circ}}\)

Therefore,

\(\displaystyle{A}\approx{139}^{{\circ}}\)

\(\displaystyle{B}\approx{14}^{{\circ}}\)

\(\displaystyle{C}\approx{26}^{{\circ}}\)

The law of cosines is given by

\(\displaystyle{\cos{{A}}}={\frac{{{b}^{{{2}}}+{c}^{{{2}}}-{a}^{{{2}}}}}{{{2}{b}{c}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{a}^{{{2}}}+{c}^{{{2}}}-{b}^{{{2}}}}}{{{2}{a}{c}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{a}^{{{2}}}+{b}^{{{2}}}-{c}^{{{2}}}}}{{{2}{a}{b}}}}\)

Step 2

Given

\(\displaystyle{a}={5},\ {b}={5},\ {c}={5}\)

Using law of cosines,

\(\displaystyle{\cos{{A}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{25}+{25}-{25}}}{{{50}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle{A}={{\cos}^{{-{1}}}{\left({\frac{{{1}}}{{{2}}}}\right)}}\)

\(\displaystyle\Rightarrow{A}={60}^{{\circ}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{B}={60}^{{\circ}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{\left({5}\right)}^{{{2}}}+{\left({5}\right)}^{{{2}}}-{\left({5}\right)}^{{{2}}}}}{{{2}{\left({5}\right)}{\left({5}\right)}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{1}}}{{{2}}}}\)

\(\displaystyle\Rightarrow{C}={60}^{{\circ}}\)

Therefore,

\(\displaystyle{A}={60}^{{\circ}}\)

\(\displaystyle{B}={60}^{{\circ}}\)

\(\displaystyle{C}={60}^{{\circ}}\)

Step 3

Given,

\(\displaystyle{a}={66},\ {b}={25},\ {c}={45}\)

Using law of cosines,

\(\displaystyle{\cos{{A}}}={\frac{{{\left({25}\right)}^{{{2}}}+{\left({45}\right)}^{{{2}}}-{\left({66}\right)}^{{{2}}}}}{{{2}{\left({25}\right)}{\left({45}\right)}}}}\)

\(\displaystyle{\cos{{A}}}={\frac{{{625}+{2025}-{4356}}}{{{2250}}}}\)

\(\displaystyle{\cos{{A}}}=-{\frac{{{853}}}{{{1125}}}}\)

\(\displaystyle{A}={{\cos}^{{-{1}}}{\left(-{\frac{{{853}}}{{{1125}}}}\right)}}\)

\(\displaystyle\Rightarrow{A}\approx{139}^{{\circ}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{\left({66}\right)}^{{{2}}}+{\left({45}\right)}^{{{2}}}-{\left({25}\right)}^{{{2}}}}}{{{2}{\left({66}\right)}{\left({45}\right)}}}}\)

\(\displaystyle{\cos{{B}}}={\frac{{{1439}}}{{{1485}}}}\)

\(\displaystyle\Rightarrow{B}\approx{14}^{{\circ}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{\left({66}\right)}^{{{2}}}+{\left({25}\right)}^{{{2}}}-{\left({45}\right)}^{{{2}}}}}{{{2}{\left({66}\right)}{\left({25}\right)}}}}\)

\(\displaystyle{\cos{{C}}}={\frac{{{739}}}{{{825}}}}\)

\(\displaystyle\Rightarrow{C}\approx{26}^{{\circ}}\)

Therefore,

\(\displaystyle{A}\approx{139}^{{\circ}}\)

\(\displaystyle{B}\approx{14}^{{\circ}}\)

\(\displaystyle{C}\approx{26}^{{\circ}}\)