236. Write the number $z=4\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)$ in algebraic and exponential forms.
Solution. Since the argument $\varphi$ of the given number is $4 \pi / 3$, the number $z$ corresponds to a point on the complex plane located in the III quadrant. Using the reduction formulas, we find

$$
\begin{aligned}
\cos \frac{4 \pi}{3}=\cos \left(\pi+\frac{\pi}{3}\right)= & -\cos \frac{\pi}{3}=-\frac{1}{2}, \sin \frac{4 \pi}{3}=\sin \left(\pi+\frac{\pi}{3}\right)= \\
= & -\sin \frac{\pi}{3}=-\frac{\sqrt{3}}{2}
\end{aligned}
$$

Substitute the obtained values into the trigonometric form of the number and expand the brackets:

$$
\begin{gathered}
z=4\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)=4\left(-\frac{1}{2}+i\left(-\frac{\sqrt{3}}{2}\right)\right)=-4 \cdot \frac{1}{2}+ \\
+4 i\left(-\frac{\sqrt{3}}{2}\right)=-2-2 i \sqrt{3}
\end{gathered}
$$

Thus, the algebraic form of the given number is $z=-2-$ $-2 i \sqrt{3}$, and the exponential form is $z=4 e^{\frac{4 \pi}{3} i}$.

237-242. Write the complex numbers in trigonometric and exponential forms:

$\begin{array}{ll}\text { 237. } z=5 i . & \text { 238. } z=-6 . \quad \text { 239. } z=-2-2 i \text {. }\end{array}$