De Moivre’s theorem
Raise a complex number in polar form to an integer power.
De Moivre’s theorem is one of 3 complex numbers formulas in the precalculus section of this library, and it is used at high school · university level.
Why de moivre’s theorem works
Multiplying two complex numbers in polar form multiplies their moduli and adds their arguments. Raising a number to the power n is multiplying it by itself n times, so the modulus gets multiplied by itself n times while the argument gets added to itself n times. One becomes a power and the other becomes a product.
What each symbol means
$r$ is modulus, $\theta$ an argument, and $n$ an integer.
De Moivre’s theorem: when it holds
Use radians consistently when calculating the angle numerically.
When it stops applying
For a fractional exponent it reports only one of several answers. A number has n distinct nth roots spread evenly around a circle, and the theorem as written hands back just the one with the smallest angle, so the others must be found by adding 2πk/n.
De Moivre’s theorem: a worked example
$(\cos\frac\pi3+i\sin\frac\pi3)^3=\cos\pi+i\sin\pi=-1$.
The mistake to avoid
What people do: Students distribute the exponent across the bracket, writing cos³θ + i sin³θ.
Why it goes wrong: Cubing a sum is nothing like cubing each piece. At θ = π/3 the true value of the cube is −1, while the term-by-term version gives 0.125 + 0.650i, which is not even close.
Do this instead: Multiply the angle instead of powering the functions: the cube of cos(π/3) + i sin(π/3) is cos π + i sin π, which equals −1 exactly.
De Moivre’s theorem: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is modulus, $\theta$ an argument, and $n$ an integer.
- Check the conditions before substituting. Use radians consistently when calculating the angle numerically.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most precalculus slips.
Where this formula fits
- Subject
- Precalculus formulas — 10 entries in this library
- Topic
- Complex numbers
- Level
- High school · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where de moivre’s theorem comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained — the lesson behind this formula: use the formula and interpret its discriminant.
- Precalculus Calculator — check your substitution and the value it produces.
- Study precalculus — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 10 precalculus formulas — the full grouped reference, or the complete formula library.
Questions about de moivre’s theorem
Can the exponent be negative?
Yes. A negative n gives r raised to that negative power, which is a reciprocal, and an angle turned the other way. That matches the fact that dividing complex numbers subtracts their arguments.
How do I find all three cube roots of a number?
Take the cube root of the modulus, divide the argument by 3, then add 120 degrees twice to get the other two. The three roots always sit at the corners of an equilateral triangle centered on the origin.
Why is the modulus raised to a power while the angle is multiplied?
Because they play different roles in a product. Lengths multiply, and multiplying the same length n times is exponentiation, while directions add, and adding the same angle n times is plain multiplication.
What is it used for besides powers?
It derives multiple-angle identities in one line. Expanding the square of cos θ + i sin θ and matching real and imaginary parts hands you the double-angle formulas for both sine and cosine at once.