i know to solve the question if it is given a+ib . but for this kind of question i can't solve it because it is only given 'ib' . for this type of question i am stuck on how to find the angle . could you help me to solve it using De Moivre's Theorem ? your help would be greatly appreciated . (=
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$\begingroup$You have that $\mathrm{i} = \exp(\mathrm{i} \pi / 2)$, and you are all set.
$\endgroup$ $\begingroup$Since $-27i=(3i)^3$ it follows that $z_0=3i$ is one of such roots, by taking the another cubic roots of the unity, $\omega \neq 1\neq\omega^2$ you get the another two cubic roots: $3i\omega$ and $3i\omega^2$.
$\endgroup$ $\begingroup$Using Euler reaction, directly we have \frac13 rotation in Argand diagram..
$$ 3 ( \cos \pi/3, \sin \pi/3), 3 ( \cos 2\pi/3, \sin 2 \pi/3),(0,-27 i). $$
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