Translations:Sinusoidal motion/9/en

The function $e^{i\omega t}$ represents the rotating vector shown in Figure 2a. The angular frequency is $\omega$, which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is $f{\rm {=}}\omega {\rm {/2}}\pi$. As t increases, the vector $e^{i\omega t}$ 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 $e^{i\omega t}$ for fixed $\omega$ and t represents a vector whose projection on the x-axis is ${\rm {\ cos\ }}\omega t$ and whose projection on the y-axis is ${\rm {\ sin\ }}\omega t$. The angle of this vector is $\omega t$, and the length of this vector is one. The (x,y)-plane is called the complex z-plane, where $z{\rm {=}}x{\rm {+}}iy$. 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.