Sum of two Angles Identities

The sum of two angles identities are trigonometric identities that express the cosine and sine of the sum of two angles in terms of the trigonometry their individual angles.

The identities can be derived a number of ways.

  • Using the right triangle definitions of the trigonometric functions[1]
  • Using the circle definitions of the trigonometric functions[2]
  • Using the trigonometry of the complex plane[3]

The sum of two angle identities can be visualized on the unit circle as shown below. The point on the unit circle corresponding to the angle has a horizontal component equal to and a vertical coordinate equal to . Both of these lengths can be equated to the trigonometry of the individual angles. Read more…

Sum of two angles identities on the unit circle.


  1. Derive Sum of Two Angles Identities (Right Triangle)
    Wumbo (internal)
  2. Derive Sum of Two Angles Identities (Unit Circle)
    Wumbo (internal)
  3. Derive Sum of Two Angles Identities (Complex Plane)
    Wumbo (internal)