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How do sine and cosine functions model repeating patterns?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

Sine and cosine functions model repeating patterns because they are periodic functions, meaning their values recur at regular intervals. When defined as the projections of a point moving along a unit circle, these functions repeat their values every $2 π$ radians (or 360°).123

This periodicity makes them essential for describing periodic motion, such as the oscillations of a pendulum, the vibrations of a tuning fork, or the propagation of electromagnetic waves. In these applications, the position of a particle can be expressed using the formula $y = r \sin (ω t + ϵ)$ , where $r$ is the amplitude, $ω$ is the angular speed, and $ϵ$ is the initial phase angle.134

Furthermore, Joseph Fourier demonstrated that almost any periodic function can be represented as an infinite sum of sine and cosine functions, known as a Fourier series. This mathematical foundation allows scientists to analyze complex phenomena ranging from acoustics and radio transmission to earthquake patterns.3