# Use the Law of Cosines to solve the triangles. Round

Use the Law of Cosines to solve the triangles. Round lengths to the nearest tenth and angle measures to the nearest degree.
$$\displaystyle{a}={4},\ {b}={6},\ {c}={9}$$
$$\displaystyle{a}={4},\ {b}={7},\ {c}={6}$$

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Step 1
For a triangle with sides a, b, c and angles A, B, C the law of consines is defined as:
$$\displaystyle{c}^{{{2}}}={a}^{{{2}}}+{b}^{{{2}}}-{2}{a}{b}\times{\cos{{\left({C}\right)}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({C}\right)}}}={\frac{{{a}^{{{2}}}+{b}^{{{2}}}-{c}^{{{2}}}}}{{{2}{a}{b}}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({C}\right)}}}={\frac{{{4}^{{{2}}}+{6}^{{{2}}}-{9}^{{{2}}}}}{{{2}\times{4}\times{6}}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({C}\right)}}}=-{0.60416}$$
$$\displaystyle\Rightarrow{C}={{\cos}^{{-{1}}}{\left(-{0.60416}\right)}}$$ (Because $$\displaystyle{\cos{{\left({C}\right)}}}$$ is negative, C is an obtuse angle.)
$$\displaystyle\Rightarrow\angle{C}={127.17}^{{\circ}}\approx{127}^{{\circ}}$$
Step 2
Now we will use law ofsines to find angle A.
$$\displaystyle{\frac{{{\sin{{\left({A}\right)}}}}}{{{a}}}}={\frac{{{\sin{{\left({C}\right)}}}}}{{{c}}}}$$
$$\displaystyle\Rightarrow{\sin{{\left({A}\right)}}}={\frac{{{4}{\sin{{\left({127}^{{\circ}}\right)}}}}}{{{9}}}}$$
$$\displaystyle\Rightarrow{\sin{{\left({A}\right)}}}={0.35494}$$
$$\displaystyle\Rightarrow{A}={\arcsin{{\left({0.35494}\right)}}}$$
$$\displaystyle\Rightarrow\angle{A}={20.7898}^{{\circ}}\approx{21}^{{\circ}}$$
Now $$\displaystyle\angle{B}-{180}^{{\circ}}-\angle{A}-\angle{C}$$
$$\displaystyle={180}^{{\circ}}-{127}^{{\circ}}-{21}^{{\circ}}$$
$$\displaystyle\angle{B}={32}^{{\circ}}$$
Step 3
2nd Triangle: $$\displaystyle{a}={4},\ {b}={7},\ {c}={6}$$
First we will find angle B using law of cosines.
$$\displaystyle{b}^{{{2}}}={a}^{{{2}}}+{c}^{{{2}}}-{2}{a}{c}\times{\cos{{\left({B}\right)}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({B}\right)}}}={\frac{{{a}^{{{2}}}+{c}^{{{2}}}-{b}^{{{2}}}}}{{{2}{a}{c}}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({B}\right)}}}={\frac{{{4}^{{{2}}}+{6}^{{{2}}}-{7}^{{{2}}}}}{{{2}\times{4}\times{6}}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({B}\right)}}}={0.0625}$$
$$\displaystyle\Rightarrow{B}={{\cos}^{{-{1}}}{\left({0.0625}\right)}}$$
$$\displaystyle\Rightarrow\angle{B}={86.4164}^{{\circ}}\approx{86}^{{\circ}}$$
Step 4
Now we will find A by using law of sines.
$$\displaystyle{\frac{{{a}}}{{{\sin{{\left({A}\right)}}}}}}={\frac{{{b}}}{{{\sin{{\left({B}\right)}}}}}}$$
$$\displaystyle{\sin{{\left({A}\right)}}}={\frac{{{a}{\sin{{\left({B}\right)}}}}}{{{b}}}}$$
$$\displaystyle{\sin{{\left({A}\right)}}}={\frac{{{4}{\sin{{\left({86}^{{\circ}}\right)}}}}}{{{7}}}}$$
$$\displaystyle{\sin{{\left({A}\right)}}}={0.57003}$$
$$\displaystyle{A}={\arcsin{{\left({0.57003}\right)}}}$$
$$\displaystyle\angle{A}={34.7528}^{{\circ}}\approx{35}^{{\circ}}$$
Now $$\displaystyle\angle{C}={180}^{{\circ}}-\angle{B}-\angle{A}$$
$$\displaystyle={180}^{{\circ}}-{86}^{{\circ}}-{35}^{{\circ}}={59}^{{\circ}}$$
$$\displaystyle\angle{C}={59}^{{\circ}}$$