Task B-4.5. Let $z=-\cos \frac{9 \pi}{8}+i \sin \frac{7 \pi}{8}$ be a complex number. Determine the smallest natural number $n$ such that the real part of the number $z^{n}$ is 0.
Solution. Let's write the number $z=-\cos \frac{9 \pi}{8}+i \sin \frac{7 \pi}{8}$ in trigonometric form:

$$
z=\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}
$$

Then $z^{n}=\cos \frac{n \pi}{8}+i \sin \frac{n \pi}{8}$.

Equate the real part to zero: $\cos \frac{n \pi}{8}=0$.

We get $\frac{n \pi}{8}=\frac{\pi}{2}+k \pi, k \in \mathbb{Z}$. The smallest natural number is $n=4$.