Roots of Complex Number/Examples/Cube Roots of -1+i

Example of Roots of Complex Number: Corollary
The complex cube roots of $-1 + i$ are given by:
 * $\paren {-1 + i}^{1/3} = \set {2^{1/6} \paren {\cos \dfrac \pi 4 + i \sin \dfrac \pi 4}, 2^{1/6} \paren {\cos \dfrac {11 \pi} {12} + i \sin \dfrac {11 \pi} {12} }, 2^{1/6} \paren {\cos \dfrac {19 \pi} {12} + i \sin \dfrac {19 \pi} {12} }}$

Proof

 * Complex Cube Roots of -1+i.png

Let $z^3 = -1 + i$.

We have that:
 * $z^3 = \sqrt 2 \paren {\map \cos {\dfrac {3 \pi} 4 + 2 k \pi} + i \, \map \sin {\dfrac {3 \pi} 4 + 2 k \pi} }$

Let $z = r \cis \theta$.

Then: