Converting Rectangular Form to Polar Form

To convert a complex number from rectangular form (a + bi) to polar form (r(cosθ + isinθ)), where a and b are real numbers and i is the imaginary unit, you can follow these steps:

Step 1: Identify the Real and Imaginary Parts
- Given a complex number in rectangular form, identify the real part (a) and the imaginary part (b).

Step 2: Find the Absolute Value (Magnitude)
- Calculate the absolute value (magnitude) of the complex number using the formula: r = √(a^2 + b^2).
- Square each component (a and b), sum them, and then take the square root to find the magnitude (r).

Step 3: Find the Angle (Argument)
- Calculate the angle (argument) of the complex number using the formula: θ = tan^(-1)(b/a).
- Divide the imaginary part (b) by the real part (a) and take the inverse tangent (tan^(-1)) of the result to find the angle (θ).

Step 4: Write the Polar Form
- Write the complex number in polar form using the magnitude (r) and angle (θ) obtained in the previous steps.
- The polar form is written as r(cosθ + isinθ), where r is the magnitude and θ is the angle.

Example:
Let's convert the complex number -2 + 2i to polar form.

1. Identify the real and imaginary parts:
- Real part (a) = -2
- Imaginary part (b) = 2

2. Find the absolute value (magnitude):
- r = √((-2)^2 + 2^2) = √(4 + 4) = √8 = 2√2

3. Find the angle (argument):
- θ = tan^(-1)(2/(-2)) = tan^(-1)(-1) = -π/4 (or -45°)

4. Write the polar form:
- Polar form = 2√2(cos(-π/4) + isin(-π/4))

Therefore, the rectangular form -2 + 2i is equivalent to the polar form 2√2(cos(-π/4) + isin(-π/4)).

Note: Make sure to use the correct signs for the angle based on the quadrant in which the complex number lies.