De Moivre’s theorem & powers
De Moivre’s theorem says (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ), which reduces any power to two arithmetic operations. Beyond computation it is a machine for trigonometric identities: expanding (cos θ + i sin θ)³ with the binomial theorem and comparing real parts produces the triple-angle formula in three lines.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert to polar formPowers are hopeless in Cartesian beyond the square; polar makes them immediate.
- Raise the modulus and multiply the argumentzⁿ = rⁿe^(inθ). That is the whole theorem.
- Reduce the argument mod 2πA large n gives a large angle; subtract multiples of 2π before evaluating.
- Use it for identitiesExpand (cos θ + i sin θ)ⁿ binomially and equate real and imaginary parts with cos nθ and sin nθ.
Worked example
Compute (1 + i)⁸.
- Polar form: 1 + i = √2 e^(iπ/4).
- By De Moivre: (√2)⁸ e^(i·8π/4) = 16 e^(i2π).
- e^(i2π) = 1, a full turn back to the start.
- So the result is 16.
Answer. (1 + i)⁸ = 16 — a real number, which is far from obvious in Cartesian form.
Where marks get dropped
These are the specific errors that cost credit on de moivre’s theorem & powers questions — QED's rubric penalises each of them separately.
- Raising the modulus to the wrong power, or forgetting to raise it at all. Both r and θ transform.
- Applying De Moivre with a non-unit modulus while omitting rⁿ.
- Assuming the theorem gives ALL nth roots when n is fractional. For roots you must add 2πk before dividing.
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De Moivre’s theorem & powers — frequently asked questions
How do I derive cos 3θ with De Moivre?
Expand (cos θ + i sin θ)³ = cos³θ + 3i cos²θ sin θ − 3cos θ sin²θ − i sin³θ. Equating real parts with cos 3θ gives cos³θ − 3cos θ sin²θ = 4cos³θ − 3cos θ.
Does it work for negative n?
Yes: z⁻ⁿ = r⁻ⁿe^(−inθ). The formula holds for every integer exponent.
Why is it more than a computational shortcut?
It links algebra to geometry: raising to the nth power scales the modulus and multiplies the rotation, which is what makes roots of unity sit at the vertices of a regular polygon.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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