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>B</mi></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mi>sin</mi><mrow><mo>(</mo><mn>35</mn><mo>)</mo></mrow></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>35</mn><mo>)</mo></mrow></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.57357643</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>12</mn></mstyle></math> .

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>11</mn></mstyle></math> .

Simplify both sides of the equation.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>11</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Multiply <math><mstyle displaystyle="true"><mn>11</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.04779803</mn></mstyle></math> .

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

Evaluate <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>0.52577839</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>31.72065955</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

The solution to the equation <math><mstyle displaystyle="true"><mi>B</mi><mo>=</mo><mn>31.72065955</mn></mstyle></math> .

Exclude the invalid angle.

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>35</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>31.72065955</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>66.72065955</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>66.72065955</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

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>c</mi></mstyle></math> .

Factor each term.

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>113.27934044</mn><mo>)</mo></mrow></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>35</mn><mo>)</mo></mrow></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.57357643</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>12</mn></mstyle></math> .

Solve for <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> .

Multiply each term by <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and simplify.

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>0.91858894</mn></mrow><mrow><mi>c</mi></mrow></mfrac><mo>=</mo><mn>0.04779803</mn></mstyle></math> by <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as <math><mstyle displaystyle="true"><mn>0.04779803</mn><mi>c</mi><mo>=</mo><mn>0.91858894</mn></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>0.04779803</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>0.04779803</mn><mi>c</mi><mo>=</mo><mn>0.91858894</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.04779803</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>0.04779803</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.91858894</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.04779803</mn></mstyle></math> .

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

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