De Moivre Formula with Worked Examples

Key Points

• The de moivre formula is $(\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta$, which NIST's DLMF prints for any integer $n$.

• With a modulus included, a complex number $r(\cos\theta+i\sin\theta)$ raised to the power $n$ gives $r^n(\cos n\theta + i\sin n\theta)$.

• Raising to a power multiplies the angle and raises the modulus to that power, which turns a hard multiplication into simple arithmetic.

• Every non-zero complex number has exactly $n$ distinct $n$th roots, found by adding $\frac{2k\pi}{n}$ to the angle for $k = 0$ to $n-1$.

• Expanding the left side for $n = 2$ and comparing parts produces the double-angle identities, so the formula generates trigonometric results.

What Is the De Moivre Formula?

The de moivre formula connects powers of a complex number to multiples of its angle. NIST's Digital Library of Mathematical Functions prints it, under the heading DeMoivre's Theorem, as:

$$\cos n\theta + i \sin n\theta = (\cos\theta + i\sin\theta)^n, \qquad n \in \mathbb{Z}$$

Including a modulus gives the working form for any complex number:

$$\left[r(\cos\theta+i\sin\theta)\right]^n = r^n(\cos n\theta + i\sin n\theta)$$

Why Powers Become Multiples of the Angle

Multiplying two complex numbers in polar form multiplies their moduli and adds their arguments. That single rule is the whole engine.

So squaring a number multiplies its modulus by itself and adds its argument to itself, giving $r^2$ and $2\theta$. Cubing repeats the step, giving $r^3$ and $3\theta$, and the pattern continues for any integer power.

The exponential form makes it immediate. NIST's DLMF gives $e^{i\theta} = \cos\theta + i\sin\theta$, so the left side is $(re^{i\theta})^n = r^n e^{in\theta}$, which is the formula written another way.

Variable Key

Examples of the De Moivre Formula

Example 1

Evaluate $(\cos 30^\circ + i\sin 30^\circ)^3$.

$$= \cos 90^\circ + i\sin 90^\circ = 0 + i = i$$

Final answer: $i$

Example 2

Evaluate $(1+i)^8$, treating the modulus as 1.

Writing $1+i$ in polar form gives an argument of $45^\circ$, so multiplying the angle gives $8 \times 45 = 360^\circ$, and $\cos 360^\circ + i\sin 360^\circ = 1$.

That answer of 1 cannot be right. The number $1+i$ has modulus $\sqrt{2}$, which is greater than 1, so its eighth power must be much larger than 1 in size.

The modulus has to be raised too:

$$r = \sqrt{2}, \qquad r^8 = (\sqrt{2})^8 = 2^4 = 16$$ $$(1+i)^8 = 16(\cos 360^\circ + i \sin 360^\circ) = 16$$

Final answer: 16. The angle part was right; the modulus was the omission.

Example 3

Find all three cube roots of 8.

Write $8$ as $8(\cos 0^\circ + i\sin 0^\circ)$. The roots take modulus $8^{1/3} = 2$ and angles $\frac{0+360k}{3}$ for $k = 0,1,2$:

$$k=0: \ 2(\cos 0 + i\sin 0) = 2$$ $$k=1: \ 2(\cos 120^\circ + i\sin 120^\circ) = -1 + i\sqrt{3}$$ $$k=2: \ 2(\cos 240^\circ + i\sin 240^\circ) = -1 - i\sqrt{3}$$

Final answer: $2$, $-1+i\sqrt{3}$ and $-1-i\sqrt{3}$.

Only one is real, which is why the cube root of 8 is usually quoted as 2 alone.

Example 4

Use the formula with $n = 2$ to derive the double-angle identities.

The formula gives $(\cos\theta+i\sin\theta)^2 = \cos 2\theta + i\sin 2\theta$. Expanding the left side directly:

$$\cos^2\theta + 2i\sin\theta\cos\theta + i^2\sin^2\theta = (\cos^2\theta - \sin^2\theta) + i(2\sin\theta\cos\theta)$$

Comparing real and imaginary parts:

Final answer: $\cos 2\theta = \cos^2\theta - \sin^2\theta$ and $\sin 2\theta = 2\sin\theta\cos\theta$.

Taking $n = 3$ produces the triple-angle identities the same way, which is how the whole family is generated.

Example 5

Evaluate $(\cos 20^\circ + i \sin 20^\circ)^{-3}$.

Negative powers are allowed, since the DLMF states the result for any integer $n$:

$$= \cos(-60^\circ) + i\sin(-60^\circ) = \frac{1}{2} - i\frac{\sqrt{3}}{2}$$

Final answer: $\frac{1}{2} - i\frac{\sqrt{3}}{2}$

Powers, Roots and the Polar Form Compared

The second row is the one that catches people. A power has a single value, but an $n$th root has $n$ of them, spaced equally around a circle of radius $r^{1/n}$.

De Moivre, Where Things Go Sideways

Mistake 1: Forgetting to raise the modulus

Where it slips in: The memorable form of the formula has no $r$ in it, because it is stated for a number already on the unit circle.

Don't do this: $(1+i)^8 = \cos 360^\circ + i\sin 360^\circ = 1$

The correct way: Use $r^n(\cos n\theta + i\sin n\theta)$. Check the modulus first: if it is not 1, it must be raised to the power as well.

Students who compute $r$ and $\theta$ as two separate labelled quantities before touching the power never drop the modulus, because it is already written down waiting to be used. Reading the bare unit-circle form as the general formula is what causes the omission.

Mistake 2: Finding only one nth root

Where it slips in: The formula returns an expression, and one substitution produces one answer.

Don't do this: Report the cube root of 8 as just 2.

The correct way: Step $k$ through $0$ to $n-1$. There are exactly $n$ distinct $n$th roots, equally spaced by $\frac{360^\circ}{n}$ around the circle.

Mistake 3: Mixing degrees and radians

Where it slips in: The angle gets multiplied by $n$ in one unit and evaluated in another.

Don't do this: Multiply $\frac{\pi}{6}$ by 3 and then read the cosine of 90 degrees.

The correct way: Pick one unit and keep it throughout. Multiplying the angle is safe in either, but the final evaluation must match the unit used.

Bottom Line

• The de moivre formula is $(\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta$, for any integer $n$.

• Including the modulus gives $r^n(\cos n\theta+i\sin n\theta)$, and forgetting the $r^n$ is the dominant error.

• It follows from the rule that multiplying complex numbers adds arguments and multiplies moduli.

• An $n$th root has $n$ distinct values, found by stepping $k$ from $0$ to $n-1$.

• To work through complex numbers with a teacher, explore a trigonometry tutor or math classes online.

Where to Go From Here

Evaluate $(\sqrt{3}+i)^6$ by converting to polar form first. Then find all four fourth roots of 16 and plot them to confirm they are equally spaced.

Want a live Bhanzu trainer to walk through more complex number problems? Book a free demo class.

Read More

• Argand Plane

• Modulus of a Complex Number

• Polar Coordinates

• Polar to Rectangular

• Trigonometric Chart