Derive the Trigonometric Identities using the geometry of the Complex Plane

This examples demonstrates how to derive the trigonometric identities using the geometry of the complex plane.

Background

Before proceeding it is important to understand both Euler’s Formula and the geometry of algebra in the complex plane.

Identities

Derive Sum of Two Angles Identities (Complex Plane)

Derive Difference of Two Angles Identities (Complex Plane)

  1. Start by placing two right-triangles of angles and next to eachother so that their angles sum together.

    Sum of Two Angles Identity Goal

    There are a couple of choices of how to relate these two right-triangles together. Note in this choice, the right triangle defined by is scaled by . Ordinary with a hypotenuse of length the length of the adjacent side would be and the length of the opposite side would be , but since the hypotenuse is , they are both scaled by the value of .

Derive Double Angles Identities (Complex Plane)

Sum of Two Complex Numbers

Drum roll please. If you have been following along and trying things out for yourself this is a moment worthy of reflection, the “why should I care moment”, or possible for the unconvinced skeptics jumping ahead this is the “why you might care” moment for the exponential function and complex inputs. The sum of two angles multiplied by the constant returns the complex coordinate on the unit circl for the “sum of the two angles”.

By comparison, the alternative strategy for this type of mathematics would be to store the information of a point in a vector such as .

Where the sum of two angles identities are given in the form below:

Derive Half Angle Identities (Complex Plane)