, ,

The Law of Sines produces an ambiguous angle result. This means that there are 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 .

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Multiply both sides of the equation by .

Simplify both sides of the equation.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify .

Cancel the common factor of .

Factor out of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Combine and .

Take the inverse sine of both sides of the equation to extract from inside the sine.

Evaluate .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.

Subtract from .

The solution to the equation .

The sum of all the angles in a triangle is degrees.

Add and .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

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 to the power of .

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Rewrite as .

Factor out of .

Factor out of .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

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 .

Factor each term.

Evaluate .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

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

Multiply each term by a factor of that will equate all the denominators. In this case, all terms need a denominator of . The expression needs to be multiplied by to make the denominator . The expression needs to be multiplied by to make the denominator .

Multiply the expression by a factor of to create the least common denominator (LCD) of .

Multiply by .

Multiply the expression by a factor of to create the least common denominator (LCD) of .

Multiply by .

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

Rewrite the equation as .

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 .

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Multiply both sides of the equation by .

Simplify both sides of the equation.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify .

Cancel the common factor of .

Factor out of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Combine and .

Take the inverse sine of both sides of the equation to extract from inside the sine.

Evaluate .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.

Subtract from .

The solution to the equation .

The sum of all the angles in a triangle is degrees.

Add and .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

Substitute the known values into the law of sines to find .

Factor each term.

Evaluate .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

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

Multiply each term by a factor of that will equate all the denominators. In this case, all terms need a denominator of . The expression needs to be multiplied by to make the denominator . The expression needs to be multiplied by to make the denominator .

Multiply the expression by a factor of to create the least common denominator (LCD) of .

Multiply by .

Multiply the expression by a factor of to create the least common denominator (LCD) of .

Multiply by .

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

Rewrite the equation as .

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

First Triangle Combination:

Second Triangle Combination:

Solve the Triangle b=8 , c=10 , B=30