Example 4 (1995 National High School Competition Question) Let the 20 vertices of a regular 20-sided polygon inscribed in the unit circle on the complex plane correspond to the complex numbers $z_{1}, z_{2}, \cdots, z_{20}$, respectively. Then the number of different points corresponding to the complex numbers $z_{1}^{1995}, z_{2}^{1995}, \cdots, z_{20}^{1995}$ is ( ).
A. 4
B. 5
C. 10
D. 20
Choose A. Reason: Let $z_{1}=\cos \theta+\mathrm{i} \sin \theta$, then $z_{k}=(\cos \theta+\mathrm{i} \sin \theta)\left[\cos \frac{2(k-1) \pi}{20}+\mathrm{i} \cdot\right.$ $\left.\sin \frac{2(k-1) \pi}{20}\right], 1 \leqslant k \leqslant 20$. From $1995=20 \times 99+15$, we get
$$
\begin{aligned}
z_{k}^{1995} & =(\cos 1995 \theta+\mathrm{i} \sin 1995 \theta)\left(\cos \frac{3 \pi}{2}+\mathrm{i} \sin \frac{3 \pi}{2}\right)^{k-1} \\
& =(\cos 1995 \theta+\mathrm{i} \sin 1995 \theta)(-\mathrm{i})^{k-1}(k=1,2, \cdots, 20) .
\end{aligned}
$$

Since $(-\mathrm{i})^{k-1}$ has 4 different values, $z_{k}^{1995}$ has a total of 4 different values.