Angle Formula Complex

Complex Number Angle Formula:

\[ \theta = \arg(z) = \arctan\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) \]

Angle θ (degrees):

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1. What is the Angle of a Complex Number?

Definition: The angle (or argument) of a complex number is the angle between the positive real axis and the line representing the complex number in the complex plane.

Purpose: This calculation is essential in complex analysis, electrical engineering, and physics for understanding the phase of complex quantities.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \theta = \arg(z) = \arctan\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) \]

Where:

\( \theta \) — Angle in degrees
\( \text{Re}(z) \) — Real part of complex number
\( \text{Im}(z) \) — Imaginary part of complex number

Explanation: The function uses atan2 which properly handles all quadrants and special cases (like when Re(z) = 0).

3. Importance of Complex Number Angle

Details: The angle represents the phase in signal processing, direction in vector analysis, and is crucial for complex number multiplication/division.

4. Using the Calculator

Tips: Enter both real and imaginary parts of your complex number. The calculator handles all cases including pure real, pure imaginary, and zero.

5. Frequently Asked Questions (FAQ)

Q1: What's the range of the angle result?
A: The calculator returns angles between -180° and +180° (or -π to +π radians).

Q2: What happens when both parts are zero?
A: The angle is undefined (0/0) - the calculator will show an empty result.

Q3: How is this different from simple arctangent?
A: atan2 considers the signs of both components to determine the correct quadrant.

Q4: What are some practical applications?
A: Used in AC circuit analysis, control systems, quantum mechanics, and computer graphics.

Q5: Can I get the result in radians?
A: Multiply the degree result by π/180 to convert to radians.