Defline the unit circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It serves as a fundamental tool in trigonometry, where any point on the unit circle can be expressed using coordinates \((\cos \theta, \sin \theta)\), with \(\theta\) being the angle formed with the positive x-axis. This circle is incredibly useful for understanding sine, cosine, and their relationships, especially when dealing with angles beyond 360 degrees. In practical terms, the unit circle helps in solving problems involving periodic functions. For example, when you're working with waves or oscillations, the circular motion represented by the unit circle can help determine the height of these waves at any point in time. This quality makes it invaluable in physics, engineering, and even music, where sound waves have sine wave characteristics and can be represented using the unit circle's properties.