Derive the Trigonometric Identities (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

  • Sum of Two Angles Identities
  • Difference of Two Angles Identities
  • Double Angle Identities
  • Half Angle Identities

Derive Sum of Two Angles Identities (Complex Plane)

This example demonstrates how the sum of two angles identities, shown below, can be derived using the properties of complex numbers.

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:

Steps

  1. Since the magnitude of the complex number doesn’t matter we can start with two complex numbers expressed along the complex unit circle.
  2. Test step two.

Derive Difference of Two Angles Identities (Complex Plane)

TODO:

Derive Double Angles Identities (Complex Plane)

Derive Half Angle Identities (Complex Plane)

TODO:

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