Polar Coordinates and De Moivre's Theorem

Precalculus topicsConic SectionsPolar Coordinates and De Moivre's TheoremVectors Matrices and Determinants

Polar coordinates

Polar to rectangular

x = r \cos θ y = r \sin θ

Rectangular to polar

r^{2} = x^{2} + y^{2} \tan θ = \frac{y}{x}

Distance between two points

d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})}

Polar form

z = a + b i = r (\cos θ + i \sin θ) = r cis θ = r e^{i θ}

Multiply

z_{1} z_{2} = r_{1} r_{2} cis (θ_{1} + θ_{2})

Divide

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} cis (θ_{1} - θ_{2})

De Moivre's theorem

z^{n} = r^{n} (\cos n θ + i \sin n θ)

nth roots

\sqrt[n]{z} = \sqrt[n]{r} cis \frac{θ + 2 π k}{n}

k = 0, 1, …, n − 1

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