Answer
The fourth roots of 1 are:
$cos~0^{\circ}+i~sin~0^{\circ}$
$cos~90^{\circ}+i~sin~90^{\circ}$
$cos~180^{\circ}+i~sin~180^{\circ}$
$cos~270^{\circ}+i~sin~270^{\circ}$
We can graph the fourth roots in the complex plane:
Work Step by Step
$z = 1 + 0~i$
$z = cos~0^{\circ}+i~sin~0^{\circ}$
$r = 1$ and $\theta = 0^{\circ}$
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:
$z^{1/4} = 1^{1/4}~[cos(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})+i~sin(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(0)}{4})]$
$z^{1/4} = 1~[cos~0^{\circ}+i~sin~0^{\circ}]$
$z^{1/4} = cos~0^{\circ}+i~sin~0^{\circ}$
When k = 1:
$z^{1/4} = 1^{1/4}~[cos(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})+i~sin(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(1)}{4})]$
$z^{1/4} = 1~[cos~90^{\circ}+i~sin~90^{\circ}]$
$z^{1/4} = cos~90^{\circ}+i~sin~90^{\circ}$
When k = 2:
$z^{1/4} = 1^{1/4}~[cos(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})+i~sin(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(2)}{4})]$
$z^{1/4} = 1~[cos~180^{\circ}+i~sin~180^{\circ}]$
$z^{1/4} = cos~180^{\circ}+i~sin~180^{\circ}$
When k = 3:
$z^{1/4} = 1^{1/4}~[cos(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})+i~sin(\frac{0^{\circ}}{4}+\frac{(360^{\circ})(3)}{4})]$
$z^{1/4} = 1~[cos~270^{\circ}+i~sin~270^{\circ}]$
$z^{1/4} = cos~270^{\circ}+i~sin~270^{\circ}$
We can graph the fourth roots in the complex plane: