Roots of Complex Number/Examples/4th Roots of i

Example of Roots of Complex Number: Corollary
The complex $4$th roots of $i$ are given by:
 * $i^{1/4} = \set {b, bi, -b, -bi}$

where:
 * $b = \paren {\cos \dfrac \pi 8 + i \sin \dfrac \pi 8}$

Proof
Let $z = i$.

Then:

and so:
 * $\arg z = \frac \pi 2$

Let $b$ be defined as:

Then we have that the complex $4$th roots of unity are:
 * $1, i, -1, -i$

The result follows from Roots of Complex Number: Corollary.