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How can sine and cosine model the motion of a pendulum or spring?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

Sine and cosine functions are essential for modeling simple harmonic motion, which describes the repetitive back-and-forth movement of systems like a pendulum or a mass on a spring.12

Mathematical Modeling

When a system is displaced from its equilibrium position, it experiences a restoring force proportional to that displacement, a relationship known as Hooke’s law. Applying Newton’s second law to this motion results in a differential equation where the acceleration is proportional to the negative of the displacement.13

The solution to this equation is an oscillating function, typically expressed as:
$x = A \cos (ω t)$
In this formula:
* $x$ is the displacement at time $t$ .13
* $A$ is the maximum displacement, or amplitude.13
* $ω$ is the angular frequency, which determines how rapidly the system oscillates.13

Application to Physical Systems

Because sine and cosine are periodic functions that repeat their values every $2 π$ radians, they perfectly represent the constant time intervals of a complete vibration. For a pendulum, this model holds true when the angle of displacement is small, allowing the motion to be treated as simple harmonic. These trigonometric functions are fundamental to understanding periodic motion, enabling precise timekeeping and the analysis of complex waves through Fourier series.12453