1. What is Complex Number Angle?

The angle (or argument) of a complex number represents the direction of the vector from the origin to the point (Re, Im) in the complex plane. It is measured counterclockwise from the positive real axis.

2. How Does the Calculator Work?

The calculator uses the arctangent formula:

Where:

• \( \theta \) — Angle in radians
• \( Im \) — Imaginary part of complex number
• \( Re \) — Real part of complex number

Explanation: The calculator uses the atan2 function which automatically handles all four quadrants and special cases like division by zero.

3. Importance of Complex Number Angle

Details: The angle is crucial for representing complex numbers in polar form, analyzing AC circuits, signal processing, and solving differential equations in engineering and physics.

4. Using the Calculator

Tips: Enter both real and imaginary parts as numbers. The calculator will determine the correct quadrant and provide results in both radians and degrees.

5. Frequently Asked Questions (FAQ)

Q1: Why use atan2 instead of regular arctangent?
A: atan2 automatically determines the correct quadrant and handles cases where Re = 0, providing more accurate results.

Q2: What is the range of possible angles?
A: The angle ranges from -π to π radians (-180° to 180°).

Q3: What happens when both Re and Im are zero?
A: The angle is undefined at the origin (0,0) since there is no unique direction.

Q4: How is the angle used in polar form?
A: In polar form, a complex number is expressed as r(cosθ + i·sinθ) or re^(iθ), where r is the magnitude and θ is the angle.

Q5: What are principal values of complex angles?
A: The principal value is typically taken in the range (-π, π] radians, though angles can differ by multiples of 2π.