Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

Solve the equation.

Substitute the known values into the equation.

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>52</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>43</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1849</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>2704</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3600</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1849</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>6304</mn></mstyle></math> .

Simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>52</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>104</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4455</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>6240</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4455</mn></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>6240</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mfrac><mrow><mn>297</mn></mrow><mrow><mn>416</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

Solve the equation.

Substitute the known values into the equation.

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>43</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>52</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2704</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1849</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3600</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2704</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>5449</mn></mstyle></math> .

Simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>43</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>86</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2745</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5160</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2745</mn></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>5160</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mfrac><mrow><mn>183</mn></mrow><mrow><mn>344</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

The sum of all the angles in a triangle is <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> degrees.

Add <math><mstyle displaystyle="true"><mn>44.44341562</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>57.86088698</mn></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>C</mi></mstyle></math> to the right side of the equation.

Subtract <math><mstyle displaystyle="true"><mn>102.30430261</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>102.30430261</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

These are the results for all angles and sides for the given triangle.

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