1. A unit cycle in mathematics, especially in complex numbers and trigonometry, refers to the set of points on the complex plane that lie on the unit circle. 2. The unit circle is defined as the circle with radius 1 centered at the origin (0,0) in the coordinate plane. 3. Mathematically, the unit circle is described by the equation $$x^2 + y^2 = 1$$ where $x$ and $y$ are the coordinates of any point on the circle. 4. In terms of complex numbers, any point on the unit circle can be represented as $$z = \cos \theta + i \sin \theta$$ where $\theta$ is the angle the radius makes with the positive x-axis, and $i$ is the imaginary unit. 5. This representation is fundamental in Euler's formula: $$e^{i\theta} = \cos \theta + i \sin \theta$$ which connects complex exponentials with trigonometric functions. 6. The unit cycle is important in many areas such as signal processing, Fourier analysis, and solving polynomial equations. 7. In summary, a unit cycle is the set of all complex numbers with magnitude 1, forming a circle of radius 1 centered at the origin.