De Moivre Theorem Calculator
De Moivre's theorem gives a fast way to raise a complex number to a power. If a number has modulus r and argument theta, then its nth power has modulus r to the power n and argument n times theta. Enter the modulus, the angle in radians, and the power, and this calculator returns the new modulus and angle along with the real and imaginary parts of the result. The theorem turns repeated multiplication into a single exponentiation and angle scaling, and underlies multiple-angle trigonometric identities.
De Moivre's theorem formula
(r (cos theta + i sin theta))^n = r^n (cos(n theta) + i sin(n theta))
Result modulus = r^n
Result angle = n * theta
Real = r^n * cos(n theta)
Imaginary = r^n * sin(n theta)
The modulus is raised to the power n while the angle is simply multiplied by n. Converting back gives the rectangular real and imaginary parts.
About De Moivre's theorem
- It follows directly from the rule that multiplying complex numbers adds angles and multiplies moduli.
- Setting r = 1 keeps the result on the unit circle and isolates the angle.
- Equating real and imaginary parts yields identities like cos(2 theta) and cos(3 theta).
- It is the integer-power companion of Euler's formula e^(i theta) = cos theta + i sin theta.
- Run it in reverse with fractional n to find roots, as in the roots of unity calculator.
De Moivre's theorem: frequently asked questions
What is De Moivre's theorem?
De Moivre's theorem states that for a complex number in polar form r(cos theta + i sin theta), the nth power is r to the power n times (cos(n theta) + i sin(n theta)). It turns repeated complex multiplication into one power and one angle multiplication.
How do I use this calculator?
Enter the modulus r, the argument theta in radians, and the power n. The tool computes r raised to n, multiplies the angle by n, and returns the result both as a modulus and angle and as real and imaginary parts.
Does the power have to be a whole number?
De Moivre's theorem holds for any real power, but it gives a single principal value here. For integer powers the result is the unique nth power. For fractional powers there are multiple roots, covered by the roots of unity calculator.
How do I get the rectangular form?
After raising the modulus to the power and multiplying the angle, the real part is r^n times cos(n theta) and the imaginary part is r^n times sin(n theta). This calculator reports both.
Why is De Moivre's theorem useful?
It makes powers of complex numbers fast and is the basis for deriving multiple-angle trigonometric identities and for finding complex roots. It also connects directly to Euler's formula e to the i theta.
Official sources
- NIST Digital Library of Mathematical Functions: Trigonometric Functions: Identities.
- NIST Digital Library of Mathematical Functions: Complex Variables.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.