Derive the Trigonometric Identities (Complex Plane)

This examples demonstrates how to derive the trigonometric identities using the geometry of the complex plane. This is a very elegant way to derive the trigonometric identities, but it requires understanding of complex numbers and Euler’s Formula.

Background

Given an angle as input, Euler’s formula returns a point on the unit circle in the complex plane. The formula relates complex input to the exponential function to the trigonometric functions sine and cosine as shown in the figure below.

The formula can be visualized as a number in the complex plane on the unit circle. When given the angle and the complex constant as input, the exponential function returns a complex number on the unit circle.

The reason this formula is useful when deriving trigonometric identities is because of the properties of the exponential function.

Derive Sum of Two Angles Identities (Complex Plane)

This example derives the sum of two angles identities using the properties of complex numbers and Euler’s formula.

Steps

  1. The sum of two angles (alpha) and (beta) form a complex number on the unit circle in the complex plane. From Euler’s formula we know this complex number can be expressed using cosine and sine.

    This complex number can be visualized as a point in the complex plane, where the horizontal coordinate is equal to and the vertical coordinate is equal to .

    Sum of two angles complex unit circle.

  2. Distribute the complex constant for both angles.

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

  3. Apply Euler’s formula to both sides of the equation.

    Multiply the right-hand side of the equation.

    Substitute for on the right-hand side.

    The geometry of the expressions on the right-hand can side can be visualized by drawing the trigonometry of the two angles on the unit circle.

    Sum of two angles complex geometry showing the stretch and rotation caused by multiplying two complex numbers together.

    These lengths could be solved for, as shown in this example, but here they are the natural result of the multiplication of two complex numbers.

  4. Equate the real part and complex parts together.

    Divide both sides of the second expression by to get 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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