# 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 the right side.

Simplify the numerator.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Evaluate .

Divide by .

Multiply each term by and simplify.

Multiply each term in by .

Simplify the left side of the 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 .

Since is on the right side of the equation, switch the sides so it is on the left side of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Divide by .

Verify each of the solutions by substituting them into and solving.

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.

Add and .

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 left side.

Simplify the numerator.

Factor out of .

Apply the sine double-angle identity.

Evaluate .

Evaluate .

Simplify the numerator.

Multiply by .

Multiply by .

Simplify the right side.

Evaluate .

Divide by .

Simplify both sides of the equation.

Simplify the right side.

Evaluate .

Divide by .

Evaluate .

Multiply each term by and simplify.

Multiply each term in by .

Simplify the left side of the 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 .

Since is on the right side of the equation, switch the sides so it is on the left side of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Divide by .

Verify each of the solutions by substituting them into and solving.

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