A function \( f \) is a relation that assigns to each element \( x \) in a set \( D \) exactly one element, called \( f(x) \), in a set \( Y \). The set \( D \) is called the domain of the function.
Before we dive into the unit circle, let's observe the behavior of the standard sine wave over the interval \([-2\pi, 2\pi]\). Notice how it perfectly repeats its pattern.
Periodic Function Theorem: A function \( f \) is periodic if there exists a positive real number \( p \) such that \( f(x + p) = f(x) \) for all \( x \) in the domain of \( f \). The smallest such positive number \( p \) is called the period of \( f \).
Let's map the sine wave back to its origin: The Unit Circle. Use the slider below to change the angle \( \theta \) and see how the coordinates \( (\cos\theta, \sin\theta) \) construct the triangle.
When working with trigonometric functions in Calculus, always assume the angle is measured in radians unless explicitly stated otherwise with a degree symbol (\(^\circ\)). This ensures derivative properties hold true!