, ,

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 .

Evaluate .

Divide 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.

Multiply by .

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 .

Exclude the invalid angle.

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 .

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 .

Evaluate .

Divide by .

Solve for .

Multiply each term by and simplify.

Multiply each term in by .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Divide by .

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

Solve the Triangle A=32 , a=19 , b=12