1. What is Complex to Polar Form Conversion?

The conversion from rectangular (a + bi) to polar form (r∠θ) represents a complex number in terms of its magnitude and angle. This is particularly useful in fields like electrical engineering, physics, and signal processing.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( a \) — Real part of the complex number
• \( b \) — Imaginary part of the complex number
• \( r \) — Magnitude (always non-negative)
• \( \theta \) — Angle in radians (-π to π)

Explanation: The magnitude represents the distance from the origin in the complex plane, while the angle represents the direction from the positive real axis.

3. Importance of Polar Form

Details: Polar form simplifies multiplication and division of complex numbers, and is essential for analyzing AC circuits, wave functions, and rotational systems.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will compute the magnitude (r) and angle (θ) in radians. For a=0, special cases are handled correctly.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rectangular and polar form?
A: Rectangular form uses Cartesian coordinates (a + bi), while polar form uses magnitude and angle (r∠θ).

Q2: How do I convert back to rectangular form?
A: Use \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \).

Q3: What's the range of the angle θ?
A: The angle ranges from -π to π radians (-180° to 180°), with positive angles counter-clockwise from the positive real axis.

Q4: How is the angle calculated when a=0?
A: When a=0, θ is π/2 (90°) for b>0 or -π/2 (-90°) for b<0.

Q5: Why use atan2 instead of regular arctan?
A: atan2(b,a) automatically handles all quadrants correctly, giving the proper angle for any combination of a and b.