# 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 .

Simplify the right side.

Simplify the numerator.

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

Evaluate to get .

Divide by to get .

Evaluate to get .

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 to get .

Divide by to get .

Multiply by to get .

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 to get .

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

Add and to get .

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

Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.

Add and to get .

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 right side.

Evaluate to get .

Divide by to get .

Evaluate to get .

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 to get .

Divide by to get .

Multiply by to get .

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 to get .

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