Answer
The two square roots of $i$ are:
$cos~45^{\circ}+i~sin~45^{\circ}$
$cos~225^{\circ}+i~sin~225^{\circ}$
We can graph the two square roots in the complex plane:
Work Step by Step
$z = 0 + i$
$z = cos~90^{\circ}+i~sin~90^{\circ}$
$r = 1$ and $\theta = 90^{\circ}$
We can use this equation to find the square roots:
$z^{1/n} = r^{1/n}~[cos(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})+i~sin(\frac{\theta}{n}+\frac{360^{\circ}~k}{n})]$, where $k \in \{0, 1, 2,...,n-1\}$
When k = 0:
$z^{1/2} = 1^{1/2}~[cos(\frac{90^{\circ}}{2}+\frac{(360^{\circ})(0)}{2})+i~sin(\frac{90^{\circ}}{2}+\frac{(360^{\circ})(0)}{2})]$
$z^{1/2} = 1~[cos~45^{\circ}+i~sin~45^{\circ}]$
$z^{1/2} = cos~45^{\circ}+i~sin~45^{\circ}$
When k = 1:
$z^{1/2} = 1^{1/2}~[cos(\frac{90^{\circ}}{2}+\frac{(360^{\circ})(1)}{2})+i~sin(\frac{90^{\circ}}{2}+\frac{(360^{\circ})(1)}{2})]$
$z^{1/2} = 1~[cos~225^{\circ}+i~sin~225^{\circ}]$
$z^{1/2} = cos~225^{\circ}+i~sin~225^{\circ}$
We can graph the two square roots in the complex plane:
