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.

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>40</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>120</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>40</mn></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>47</mn><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>4800</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.68199836</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>3600</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1600</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>3273.59212829</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>5200</mn></mstyle></math> .

Evaluate the root.

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

Substitute the known values into the law of sines to find <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mi>sin</mi><mrow><mo>(</mo><mn>47</mn><mo>)</mo></mrow></mrow><mrow><mn>43.89086319</mn></mrow></mfrac></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>47</mn><mo>)</mo></mrow></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.7313537</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>43.89086319</mn></mstyle></math> .

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> .

Simplify both sides of the equation.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Multiply <math><mstyle displaystyle="true"><mn>60</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.016663</mn></mstyle></math> .

Take the inverse sine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> from inside the sine.

Evaluate <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>0.99978033</mn><mo>)</mo></mrow></mstyle></math> .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> to find the solution in the second quadrant.

Subtract <math><mstyle displaystyle="true"><mn>88.79905895</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

The solution to the equation <math><mstyle displaystyle="true"><mi>A</mi><mo>=</mo><mn>88.79905895</mn></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>88.79905895</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>47</mn></mstyle></math> .

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

Subtract <math><mstyle displaystyle="true"><mn>135.79905895</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>135.79905895</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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