De Moivre's Theorem | Moosa Academy

Multiplying two complex numbers in polar form multiplies their moduli and adds their arguments. Multiplying a number by itself, repeatedly, therefore follows a pattern so regular that it can be written down once and used for every exponent. That statement is De Moivre's theorem.

Concept Finding the pattern

Start from $z = r(\cos\theta + i\sin\theta)$ and square it. Squaring is multiplying $z$ by $z$ , so the modulus multiplies by itself and the argument adds to itself:

z^2 = r^2(\cos 2\theta + i\sin 2\theta)

z^3 = r^3(\cos 3\theta + i\sin 3\theta)

z^4 = r^4(\cos 4\theta + i\sin 4\theta)

The modulus is raised to the exponent; the angle is multiplied by it. Nothing about that depends on the exponent being 2, 3 or 4.

Theorem De Moivre's Theorem

\[ \bigl[r(\cos\theta + i\sin\theta)\bigr]^{n} = r^{n}\bigl(\cos n\theta + i\sin n\theta\bigr) \]

Each power adds one more turn of θ.

Step 1 — raise r to the power n.

Step 2 — multiply θ by n.

Step 3 — reduce the angle below 360° and evaluate the cosine and sine.

Example A fifth power

Evaluate $\bigl[2(\cos 90° + i\sin 90°)\bigr]^{5}$ .

Modulus:

2^5 = 32

Argument:

5 \times 90° = 450°

Reduce below a full turn:

450° - 360° = 90°

\cos 90° = 0

and

\sin 90° = 1

= 32(0 + i \cdot 1)

⟹ 32i

The result sits on the positive imaginary axis — the equivalent angle 90° told us that before any arithmetic was done.

Note Always reduce the angle

Multiplying θ by a large n easily produces an angle past several full turns. Subtract multiples of 360° until what remains lies between 0° and 360°, then read off the trigonometric values. The reduced angle names the same direction, so the answer is unchanged — it is only easier to evaluate.

The saving is real: $(1+i)^{8}$ by repeated multiplication is seven expansions, but in polar form it is one power and one angle.

Summary

De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
Raise the modulus to the power n and multiply the argument by n.
Subtract multiples of 360° to reduce the resulting angle before evaluating cos and sin.
It replaces repeated multiplication with a single step, which is why it is used in engineering and computing.