# Translations:Sinusoidal motion/9/en

The function represents the rotating vector shown in Figure 2a. The angular frequency is , which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is . As *t* increases, the vector 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 for fixed and *t* represents a vector whose projection on the *x*-axis is and whose projection on the *y*-axis is . The angle of this vector is , and the length of this vector is one. The (*x,y*)-plane is called the *complex z-plane*, where . 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.