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De Moivre's Theorem Calculator

Verified Calculation Engine

The online De Moivre's Theorem Calculator helps you calculate instantly and solve problems related to Complex Numbers. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.

Raises a complex number to a power using De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)).

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number
De Moivre's theorem simplifies raising complex numbers to powers

Inputs

Please enter a valid Modulus r.
Modulus of complex number
Please enter a valid Argument θ (Degrees).
Argument in degrees
Please enter a valid Power n.
Power to raise complex number to

Results

Worked Examples
Example 1: Square

Find (1 + i)² using De Moivre's theorem

Inputs:
  • modulus: 1.4142
  • argument: 45
  • power: 2
Expected Outputs:
  • resultModulus: 2
  • resultArgument: 1.5708
  • resultArgumentDegrees: 90
  • polarForm: 2(cos(90°) + i sin(90°))
  • rectangularForm: 2i
Example 2: Board Level

Find (√3 + i)³ using De Moivre's theorem

Inputs:
  • modulus: 2
  • argument: 30
  • power: 3
Expected Outputs:
  • resultModulus: 8
  • resultArgument: 1.5708
  • resultArgumentDegrees: 90
  • polarForm: 8(cos(90°) + i sin(90°))
  • rectangularForm: 8i

About this calculator

Overview

De Moivre's Theorem Calculator evaluates De Moivre's Theorem for the math / complex-numbers library. Primary input cue: Modulus r [required]: Modulus of complex number. Method anchor: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number. De Moivre's theorem simplifies raising complex nu.

When to use

Use De Moivre's Theorem Calculator when you already know the inputs and need de moivre's theorem without setting up a spreadsheet.

Inputs explained

  • Modulus r [required]: Modulus of complex number
  • Argument θ (Degrees) [required]: Argument in degrees
  • Power n [required]: Power to raise complex number to

Formula / method

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number. De Moivre's theorem simplifies raising complex numbers to powers (calculator id: de-moivres-theorem-calculator).

Worked example

Worked values: modulus = 1.4142, argument = 45, power = 2 → resultModulus = 2, resultArgument = 1.5708, resultArgumentDegrees = 90, polarForm = 2(cos(90°) + i sin(90°)), rectangularForm = 2i

Interpreting results

Cross-check with a simple numeric example before relying on extreme inputs for de moivre's theorem.

Assumptions

  • Empty required inputs are not silently replaced with zeros for de moivre's theorem. [de-moivres-theorem-calculator]
  • De Moivre's Theorem is computed only from the fields you enter on this calculator.
  • Real-number arithmetic is used unless de moivre's theorem explicitly needs integers.

Limitations

  • If multiple conventions exist for de moivre's theorem, this calculator uses the convention implemented in its engine path.
  • This page does not provide a full textbook proof for every identity related to de moivre's theorem.
  • Complex/branch cuts and undefined points are not always interactively annotated for de moivre's theorem.

Important note

Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.

How to Use This Calculator

  1. Enter the required values in the input fields.
  2. Click the Calculate button.
  3. View the computed result instantly.

Formula Used

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number. De Moivre's theorem simplifies raising complex numbers to powers (calculator id: de-moivres-theorem-calculator).

Example Calculation

Worked values: modulus = 1.4142, argument = 45, power = 2 → resultModulus = 2, resultArgument = 1.5708, resultArgumentDegrees = 90, polarForm = 2(cos(90°) + i sin(90°)), rectangularForm = 2i

Frequently Asked Questions

What is De Moivre's Theorem Calculator?

De Moivre's Theorem Calculator evaluates De Moivre's Theorem for the math / complex-numbers library. Primary input cue: Modulus r [required]: Modulus of complex number. Method anchor: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number. De Moivre's theorem simplifies raising complex nu.

How does De Moivre's Theorem Calculator work?

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) For integer n, this gives the nth power of the complex number. De Moivre's theorem simplifies raising complex numbers to powers (calculator id: de-moivres-theorem-calculator).

Why use this Math Numbers calculator?

Use De Moivre's Theorem Calculator when you already know the inputs and need de moivre's theorem without setting up a spreadsheet.