De Moivre's Theorem
Simple Explanation
De Moivre's Theorem gives a direct shortcut for raising a complex number in trigonometric form to a power n: [r(cosθ+isinθ)]ⁿ = rⁿ(cos(nθ)+isin(nθ)) — raise the modulus to the power n, and multiply the angle by n.
Why Do We Need It?
Without this theorem, raising a complex number to a high power would mean multiplying it out by itself repeatedly — De Moivre's Theorem turns that into two simple steps.
Formula
De Moivre's Theorem
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))
Raising a complex number in trigonometric form to a power n raises its modulus to the power n, and multiplies its argument by n.
- n
- — the power the complex number is being raised to
When to use it: Whenever a complex number needs to be raised to a power, especially a large one — far faster than repeated multiplication.
Worked Example
Apply De Moivre's Theorem
Use De Moivre's Theorem to find [2(cos30°+isin30°)]³.
Why Does This Work?
Multiplying two complex numbers in trigonometric form, r₁(cosα+isinα) and r₂(cosβ+isinβ), gives r₁r₂(cos(α+β)+isin(α+β)) — the moduli multiply, and the angles ADD. Raising to the nth power is just multiplying the same number by itself n times, so the modulus multiplies by itself n times (giving rⁿ) and the angle adds to itself n times (giving nθ).
Real-Life Example
Analyzing repeated signal transformations
A signal processing system applies the same complex transformation repeatedly, many times in a row.
De Moivre's Theorem computes the combined effect of many repeated applications directly, without simulating each step one at a time.
Practice
Using De Moivre's Theorem, find the modulus of [3(cos40°+isin40°)]².
HardCommon mistake
Multiplying the modulus by n instead of raising it to the power n (e.g. computing 2×3=6 instead of 2³=8) — only the ANGLE is multiplied by n; the modulus is raised to the nth power.
Quick Review
- [r(cosθ+isinθ)]ⁿ = rⁿ(cos(nθ)+isin(nθ)).
- Modulus is raised to the power n; angle is multiplied by n.
- Comes from repeatedly applying the rule for multiplying two complex numbers in trig form.