Answer
$5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right),5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right),5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right)$
Work Step by Step
Method for finding complex roots:
To find $n$ distinct complex roots for any complex number $z=r\left( \cos \theta +i\sin \theta \right)$, if $z\ne 0$, in radians we use the formula given below:
${{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +360{}^\circ \cdot k}{n} \right)+i\sin \left( \frac{\theta +360{}^\circ \cdot k}{n} \right) \right]$
Where $k=0,1,2,3,....,n-1$.
So, the cube roots of $125\left( \cos 165{}^\circ +i\sin 165{}^\circ \right)$ are
${{z}_{k}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot k}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot k}{3} \right) \right],k=0,1,2$
Therefore, we can find the three complex cube roots in the following manner:
$\begin{align}
& {{z}_{0}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 0}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 0}{3} \right) \right] \\
& =5\left( \cos \frac{165{}^\circ }{3}+i\sin \frac{165{}^\circ }{3} \right) \\
& =5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right)
\end{align}$
$\begin{align}
& {{z}_{1}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 1}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 1}{3} \right) \right] \\
& =5\left( \cos \frac{525{}^\circ }{3}+i\sin \frac{525{}^\circ }{3} \right) \\
& =5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right)
\end{align}$
$\begin{align}
& {{z}_{2}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 2}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 2}{3} \right) \right] \\
& =5\left( \cos \frac{885{}^\circ }{3}+i\sin \frac{885{}^\circ }{3} \right) \\
& =5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right)
\end{align}$
So, the cube roots are $5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right),\text{ }5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right),\text{ and }5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right)$.
