# Translations:Sinusoidal motion/9/en

The function ${\displaystyle e^{i\omega t}}$ represents the rotating vector shown in Figure 2a. The angular frequency is ${\displaystyle \omega }$, which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is ${\displaystyle f{\rm {=}}\omega {\rm {/2}}\pi }$. As t increases, the vector ${\displaystyle e^{i\omega t}}$ rotates in the counterclockwise direction. In Cartesian coordinates, we let the x-axis represent the real axis and the y-axis represent the imaginary axis (Figure 2b). Then the quantity ${\displaystyle e^{i\omega t}}$ for fixed ${\displaystyle \omega }$ and t represents a vector whose projection on the x-axis is ${\displaystyle {\rm {\ cos\ }}\omega t}$ and whose projection on the y-axis is ${\displaystyle {\rm {\ sin\ }}\omega t}$. The angle of this vector is ${\displaystyle \omega t}$, and the length of this vector is one. The (x,y)-plane is called the complex z-plane, where ${\displaystyle z{\rm {=}}x{\rm {+}}iy}$. As time t increases, this vector rotates in a counterclockwise direction, and the tip of the vector traces out a circle. Because this circle has unit radius, it is called the unit circle in this complex z-plane.