A “Proof by Picture” of the Angle Addition Formulas

Original proof by Sunny Snaith.

Original proof by Sunny Snaith.

All you need to know in order to follow the proof is:

  • The basic trig identities:
    • The center angle of the triangle is Θ.
    • The hypotenuse is 1 (a radius of the unit circle).
    • The side adjacent is cos Θ.
    • The side opposite is sin Θ.
    • The angle between the adjacent and opposite sides is a right angle.
  • That any triangle can be scaled by a factor H:
    • The angles stay the same.
    • The sides are all multiplied by H.
    • The hypotenuse in this example would become 1 × H = H.