De Moivre Theorem Calculator

Work out de moivre theorem instantly with clear inputs, formula shown and shareable results.

About De Moivre Theorem Calculator

De Moivre's theorem says (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ). It converts a repeated multiplication into a single scaling and rotation, and it is the source of the multiple-angle trigonometric identities.

Formula

De Moivre's theorem

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

Worked example

With the default values below, De Moivre Theorem Calculator returns zⁿ in rectangular form of -64 + 0i. Change any input above and every figure here is recalculated in your browser.

Inputs used

Modulus rminimum 0: 2
Argument θ: 30 °
Power n: 6

Results

zⁿ in rectangular form: -64 + 0i
New modulus rⁿ: 64.000000
New argument nθ: 180.000000 °
Argument reduced to 0°–360°: 180.000000 °

Step by step

1Values used
2De Moivre's theorem
3zⁿ in rectangular form= -64 + 0i
4New modulus rⁿ= 64.000000
5New argument nθ= 180.000000 °
6Argument reduced to 0°–360°= 180.000000 °

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For r = 2, θ = 30°, n = 6?

The modulus becomes 64 and the argument 180°, so the result is −64.

How does it give trig identities?

Expanding (cos θ + i sin θ)³ and matching real parts yields cos 3θ = 4cos³θ − 3cos θ immediately.