Task B-4.3. Determine the complex number $z^{3}$, if the imaginary part of the number $z$ is equal to $-\frac{1}{3}$ and if the argument of the number $z$ is equal to $\frac{7 \pi}{6}$.
Solution. Let $z=a+bi$, argument $z=\varphi$, then $b=-\frac{1}{3}$ and $\varphi=\frac{7 \pi}{6}$.

From $\operatorname{tg} \varphi=\frac{b}{a}$ we get $a=-\frac{\sqrt{3}}{3}$.

(1 point)

Since $|z|=\sqrt{a^{2}+b^{2}}=\frac{2}{3}$, we have

$$
z^{3}=|z|^{3}(\cos 3 \varphi+i \sin 3 \varphi)
$$

or

$$
z^{3}=\frac{8}{27}\left(\cos \frac{7 \pi}{2}+i \sin \frac{7 \pi}{2}\right)=-\frac{8}{27} i
$$