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 the geometry of Euler’s Formula for complex numbers. Namely, that when the complex constant and an angle (theta) are passed as input to the exponential function, the output is the corresponding complex number on the unit circle.

This is visualized by the complex number on the unit circle in the complex plane below.

A point on the unit circle in the complex plane.
Figure 1: Complex Unit Circle

Identities

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

Derive Sum of Two Angles Identities (Complex Plane)

This example demonstrates how the sum of two angles identities can be derived using the properties of complex numbers and Euler’s formula. The two summation identities are shown below:

Steps

  1. Observe that the angle (theta) forms the complex number on the unit circle given by Euler’s formula.

    This complex number is visualized by the figure below:

    A point on the unit circle in the complex plane.
    Figure 1: Complex Unit Circle
  2. Model the addition of the two angles (alpha) and (beta) as input to the exponential function.

    Distribute the complex constant for both angles.

  3. Apply the addition property of the exponential function to expand to right-hand side.

  4. Apply Euler’s formula to both sides of the equation where the exponential function appears.

    Multiply the right-hand side of the equation.

    Substitute for in the expression.

  5. Then, taking the last expression from the step above we can separate real parts from the complex parts of the number.

    Dividing both sides of the second expression by gives us the two summation identities.

Derive Difference of Two Angles Identities (Complex Plane)

This example demonstrates how the difference of two angles identities can be derived using the properties of complex numbers and Euler’s formula. The two difference identities are shown below:

Steps

  1. Observe that the angle (theta) forms the complex number on the unit circle given by Euler’s formula.

    This complex number is visualized by the figure below:

    A point on the unit circle in the complex plane.
    Figure 1: Complex Unit Circle
  2. Model the difference of the two angles (alpha) and (beta) as input to the exponential function.

    Distribute the complex constant for both angles.

  3. Apply the addition property of the exponential function to expand to right-hand side.

  4. Apply Euler’s formula to both sides of the equation where the exponential function appears. Note, when applying Euler’s formula for the negative angle the following symmetry can be observed.

    A point on the unit circle in the complex plane corresponding to a negative angle.
    Figure 2: Complex Unit Circle Negative Angle

    This results in the following expression.

    Multiply the right-hand side of the equation.

    Substitute for in the expression.

    Simplify and group the real and complex parts of the right-hand side.

  5. Equate the real parts of the complex number and the complex parts of the number.

    Divide both sides of the second expression by to finish deriving the two summation identities.

Derive Double Angles Identities (Complex Plane)

This example demonstrates how to derive the double angle identities using the properties of complex numbers in the complex plane. The double angle identities are shown below.

Steps

  1. Observe that the angle (theta) forms the complex number on the unit circle given by Euler’s formula.

    This complex number is visualized by the figure below:

    A point on the unit circle in the complex plane.
    Figure 1: Complex Unit Circle
  2. Model the double angle (alpha) as the input to the exponential function.

    Distribute the complex constant .

  3. Apply the addition property of the exponential function to expand to right-hand side.

  4. Apply Euler’s formula to both sides of the equation where the exponential function appears.

    Multiply the right-hand side of the equation.

    Substitute for in the expression.

  5. Equate the real parts and complex parts of the two complex numbers.

    This gives us the cosine double angle identity.

    And the sine double angle identity.

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