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 and square it. Squaring is multiplying by , so the modulus multiplies by itself and the argument adds to itself:

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 .

Modulus:

Argument:

Reduce below a full turn:

and

⟹ 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: 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.