How to Convert Polar to Exponential Form

To convert a complex number from polar form to exponential form, you use Euler's formula.

A complex number in polar form is written as z = r(cos(θ) + i sin(θ)). According to Euler's formula, the expression (cos(θ) + i sin(θ)) is equivalent to e^(iθ). Therefore, you can replace this part of the polar form with e^(iθ) to get the exponential form.

The conversion is straightforward:

• Identify the modulus r and the argument (angle) θ from the polar form.
• Substitute these values into the exponential form template.

Understanding the Forms

Let's break down the components involved in the conversion:

• Polar Form: Represents a complex number z using its distance from the origin (r, the modulus) and the angle it makes with the positive x-axis (θ, the argument).

  • z = r(cos(θ) + i sin(θ))

• Exponential Form: A more compact representation derived using Euler's formula, also using the modulus and argument.

  • z = re^(iθ)

Here's a simple comparison of the key parts:

Polar Form	Exponential Form	Meaning
r	r	Modulus (Distance from origin)
(cos(θ) + i sin(θ))	e^(iθ)	Direction (Related to the angle)
θ	θ	Argument (Angle)

The core transformation lies in the identity e^(iθ) = cos(θ) + i sin(θ), which is a direct result of Euler's formula.

Steps for Conversion

1. Start with the complex number in polar form: z = r(cos(θ) + i sin(θ))
2. Identify 'r' and 'θ': Extract the value of the modulus r and the argument θ. Ensure θ is in radians for the exponential form, although degrees are sometimes used in polar form notation (conversion is needed if necessary).
3. Apply Euler's formula: Replace the (cos(θ) + i sin(θ)) part with e^(iθ).
4. Write the exponential form: Combine r and e^(iθ) to get z = re^(iθ).

Example

Let's convert the complex number z = 2(cos(π/3) + i sin(π/3)) from polar form to exponential form.

1. Polar Form: z = 2(cos(π/3) + i sin(π/3))
2. Identify r and θ: Here, r = 2 and θ = π/3 radians.
3. Apply Euler's formula: We replace (cos(π/3) + i sin(π/3)) with e^(iπ/3).
4. Exponential Form: The exponential form is z = 2e^(iπ/3).

It's that simple! You are essentially just substituting one mathematical expression for another based on Euler's powerful identity.