De Moivre’s Theorem
In short: De Moivre’s theorem says that raising r(cos θ + i sin θ) to the power n gives rⁿ(cos nθ + i sin nθ): the modulus is raised to n and the angle is multiplied by n.
PrecalculusNumbers and OperationsN-CN.B.5
What is De Moivre's Theorem?
De Moivre’s theorem turns a heavy job into an easy one. Multiplying a complex number by itself eight times in rectangular form is tedious, but in polar form the powers act on the distance and the angle separately.
For a complex number \(z = r(\cos\theta + i\sin\theta)\) and an integer \(n\), \(z^n = r^n\big(\cos(n\theta) + i\sin(n\theta)\big)\). Geometrically, each additional power stretches the point by a factor of \(r\) and turns it through another angle \(\theta\) about the origin. The idea extends to roots by dividing the angle.
How to spot it
- A complex number in the form a + bi is raised to a high power such as 6 or 8, which calls for De Moivre’s theorem.
- An expression looks like [r(cos θ + i sin θ)]ⁿ with a numeric exponent outside.
- A question asks for a power to be left in a + bi form after simplifying an angle like 360° or 180°.
- Roots of a complex number are requested, which uses the same theorem in reverse.
The method
- Convert the number to polar form so that De Moivre’s theorem applies: find \(r\) and \(\theta\).
- Raise the modulus to the power: compute \(r^n\).
- Multiply the angle by the exponent: compute \(n\theta\), and reduce it by full turns if convenient.
- Evaluate the cosine and sine, then convert back to \(a + bi\) if asked.
De Moivre's Theorem: solved example
Problem
Use the theorem to compute \((\sqrt{3} – i)^6\).
Solution
The modulus is \(r = \sqrt{3 + 1} = 2\), and the point is in Quadrant IV with reference angle \(30^{\circ}\), so \(\theta = -30^{\circ}\).
Then \(z^6 = 2^6\big(\cos(-180^{\circ}) + i\sin(-180^{\circ})\big) = 64(-1 + 0i)\). The result is \(-64\).
Try it
Question 1
Compute \(\big[2(\cos 20^{\circ} + i\sin 20^{\circ})\big]^3\) in \(a + bi\) form.
Show the answer
\(8(\cos 60^{\circ} + i\sin 60^{\circ}) = 8\left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = 4 + 4\sqrt{3}\,i\).
Question 2
Find \((1 – i)^4\).
Show the answer
\(r = \sqrt{2}\), \(\theta = -45^{\circ}\); then \(4(\cos(-180^{\circ}) + i\sin(-180^{\circ})) = -4\).
De Moivre's Theorem vs. Polar Form of a Complex Number
Polar form is the way of writing a number with a distance and an angle. This theorem is the power rule that you apply once the number is in that form. Without the first step the second cannot start, so a typical solution always converts to polar form and only then raises to the power.
Keep going
Lessons
Worksheets and quizzes
- Polar Form and De Moivre’s Theorem – Worksheet
- Polar Form and De Moivre’s Theorem – Quiz
Related words
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