Chapter 9 - Trigonometric Identities, Models, and Complex Numbers - 9.6 Complex Numbers and De Moivre's Theorem - Exercises and Problems for Section 9.6 - Exercises and Problems - Page 392: 2

$\displaystyle e^{i \frac{3 \pi}{2}}$

The given complex number is $-i$. It can be written in the complex form $z=x+i y$ as: $z=0-i$ We see that in the complex plane, the $x$-coordinate is $0$ and the $y$-coordinate is $-1$. So, we have: $r=|z|=\sqrt {(0)^2+(-1)^2}=1$ and $\theta=\tan^{-1} (\dfrac{-1}{0})=\tan^{-1} (\infty)=\dfrac{3 \pi}{2}$ $z$ lies on the negative side of the x-axis. Therefore, we have: $z=re^{i \theta}=(1)e^{i \frac{3 \pi}{2}}=\displaystyle e^{i \frac{3 \pi}{2}}$