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Complex Number to Trigonometric Form — result sheet
Convert a complex number from rectangular real-plus-imaginary form into magnitude and angle form.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: r = √(x²+y²); θ = atan2(y,x); z = r(cos θ + i sin θ).
The modulus gives the distance from the origin in the complex plane, while atan2 preserves the correct quadrant for the argument. Both radians and degrees are shown.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Real part — minimum -1000000000; maximum 1000000000
- Imaginary part — minimum -1000000000; maximum 1000000000
Worked example
| Input | Value |
|---|---|
| Real part | 3 |
| Imaginary part | 4 |
Magnitude: 5; argument: 53.13° (0.9273 rad).
Assumptions and limits
- The input represents z = x + yi with real numeric components.
- The zero complex number has no unique argument and is rejected rather than assigned an arbitrary direction.
- The displayed angle is the principal argument from -180° through 180°; equivalent angles differ by full turns.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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