Answer
$[{cis - 22.5^{\circ},cis -112.5^{\circ},cis -202.5^{\circ}},cis -292.5^{\circ}]$
Work Step by Step
Given: $x^{4}+i=0$ or $x^{4}=-i$ $-i$ can be written in trigonometric form as: $-i=0-i=(\cos 90^{\circ}-\sin 90^{\circ})$
Absolute value of fourth root is given as $\sqrt[4] 1=1$
Now, the arguments can be given as: $k=0,1,2,3$
Roots: $(\cos 22.5^{\circ}+\sin 22.5^{\circ})$,
$(\cos 112.50^{\circ}+\sin 112.50^{\circ})$,
$(\cos 202.5^{\circ}+\sin 202.5^{\circ})$,
$(\cos 292.5^{\circ}+\sin 292.5^{\circ})$,
Solution set of the equation can be written as:
$[{cis - 22.5^{\circ},cis -112.5^{\circ},cis -202.5^{\circ}},cis -292.5^{\circ}]$
