# Trigonometry Examples

, ,

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 .

Evaluate .

Simplify .

Simplify the numerator.

Factor out of .

Apply the sine double-angle identity.

Evaluate .

Evaluate .

Simplify the numerator.

Multiply by .

Multiply by .

Divide by .

Simplify .

Evaluate .

Divide by .

Evaluate .

Solve for .

Multiply each term by and simplify.

Multiply each term in by .

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Simplify.

Multiply and .

Divide by .

Multiply by .

Rewrite the equation as .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Divide by .

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 .

Evaluate .

Simplify .

Evaluate .

Divide by .

Simplify .

Evaluate .

Divide by .

Evaluate .

Solve for .

Multiply each term by and simplify.

Multiply each term in by .

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Simplify.

Multiply and .

Divide by .

Multiply by .

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.