The Law of Sines produces an ambiguous angle result. This means that there are <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

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>30</mn><mo>)</mo></mrow></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> .

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><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> .

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><mn>16</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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.

The exact value of <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>90</mn></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>90</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>90</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>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>30</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>120</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>120</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>60</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></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><msqrt><mn>3</mn></msqrt></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> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>90</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Set up the rational expression with the same denominator over the entire equation.

Multiply each term by a factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> that will equate all the denominators. In this case, all terms need a denominator of <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> . The <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn><mi>b</mi></mrow></mfrac></mstyle></math> expression needs to be multiplied by <math><mstyle displaystyle="true"><mfrac><mrow><mn>16</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> to make the denominator <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> . The <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>32</mn></mrow></mfrac></mstyle></math> expression needs to be multiplied by <math><mstyle displaystyle="true"><mfrac><mrow><mi>b</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> to make the denominator <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> .

Multiply the expression by a factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to create the least common denominator (LCD) of <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Multiply the expression by a factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to create the least common denominator (LCD) of <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

Rewrite the equation as <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>16</mn><msqrt><mn>3</mn></msqrt></mstyle></math> .

For the second triangle, use the second possible angle value.

Solve for the second triangle.

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>30</mn><mo>)</mo></mrow></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> .

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><mo>⋅</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> .

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><mn>16</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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.

The exact value of <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>90</mn></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>90</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>90</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>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>30</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>120</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>120</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

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>60</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></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><msqrt><mn>3</mn></msqrt></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> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>90</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Set up the rational expression with the same denominator over the entire equation.

Multiply each term by a factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> that will equate all the denominators. In this case, all terms need a denominator of <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> . The <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn><mi>b</mi></mrow></mfrac></mstyle></math> expression needs to be multiplied by <math><mstyle displaystyle="true"><mfrac><mrow><mn>16</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></math> to make the denominator <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> . The <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>32</mn></mrow></mfrac></mstyle></math> expression needs to be multiplied by <math><mstyle displaystyle="true"><mfrac><mrow><mi>b</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> to make the denominator <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> .

Multiply the expression by a factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to create the least common denominator (LCD) of <math><mstyle displaystyle="true"><mn>32</mn><mi>b</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mn>16</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite the equation as <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>16</mn><msqrt><mn>3</mn></msqrt></mstyle></math> .

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

First Triangle Combination:

Second Triangle Combination:

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