Polar Form and De Moivre's Theorem
Any non-zero z can be written r(cos t + i sin t) with r = |z| and t = arg z, and (cos t + i sin t)n = cos nt + i sin nt.
What is Polar Form and De Moivre's Theorem?
Any non-zero z can be written r(cos t + i sin t) with r = |z| and t = arg z, and (cos t + i sin t)n = cos nt + i sin nt.
Key formula / rule: arg(z1 z2) = arg z1 + arg z2, modulo 2*π.
Key points
- Convert a complex number between algebraic and polar form.
- Compute the nth roots of a complex number using De Moivre's theorem.
Common exam trap
Reading the argument from the ratio y/x without checking the quadrant.
Definitions
- Term
Polar Form and De Moivre's Theorem
- Meaning
Any non-zero z can be written r(cos t + i sin t) with r = |z| and t = arg z, and (cos t + i sin t)n = cos nt + i sin nt.
- Term
Polar Form and De Moivre's Theorem — explanation
- Meaning
Polar form turns multiplication into addition of arguments, which is why powers and roots become routine. De Moivre's theorem is the engine behind nth roots, trigonometric multiple-angle identities and roots of unity.
Learning objectives
Convert a complex number between algebraic and polar form.
Compute the nth roots of a complex number using De Moivre's theorem.
Formulae
- Key point
arg(z1 z2) = arg z1 + arg z2, modulo 2*π.
- Key point
The principal argument is taken in (-π, π].
- Key point
n distinct nth roots lie equally spaced on a circle of radius r^(1/n).
Prerequisites
MAT-U1-CQ-T1-S1-C2
Common mistakes
Reading the argument from the ratio y/x without checking the quadrant.
Losing roots by taking only the principal value when extracting nth roots.
Keywords
Polar
Form
Moivre's
Theorem
Practice preview
Using De Moivre's Theorem, evaluate (cos(pi/6) + i sin(pi/6))^3.…
easy
Evaluate (cos(pi/3) + i sin(pi/3))^-2 using De Moivre's Theorem.…
medium
Calculate the value of (1 - i)^6.…
medium