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> .

Factor each term.

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>115</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>200</mn></mstyle></math> .

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

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

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>0.90630778</mn></mrow><mrow><mi>a</mi></mrow></mfrac><mo>=</mo><mn>0.00286788</mn></mstyle></math> by <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

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

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as <math><mstyle displaystyle="true"><mn>0.00286788</mn><mi>a</mi><mo>=</mo><mn>0.90630778</mn></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>0.00286788</mn><mi>a</mi><mo>=</mo><mn>0.90630778</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.90630778</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.00286788</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>115</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>35</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>150</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>150</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>b</mi></mstyle></math> .

Factor each term.

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>30</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>115</mn><mo>)</mo></mrow></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.90630778</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>316.01988143</mn></mstyle></math> .

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

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

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>b</mi></mrow></mfrac><mo>=</mo><mn>0.00286788</mn></mstyle></math> by <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> .

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

Factor <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>b</mi></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as <math><mstyle displaystyle="true"><mn>0.00286788</mn><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>0.00286788</mn><mi>b</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>0.00286788</mn></mrow></mfrac></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.00286788</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>348.68935912</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>348.68935912</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

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

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