Proof of the small angle approximation sin θ ≈ θ using the geometry and motion of a simple pendulum

Abstract

The small angle approximation sin(θ)≈θ is central to all treatments of the simple pendulum as a harmonic oscillator and is typically asserted as a result that follows from calculus. Here, however, we show that the geometry of the pendulum itself offers a route to understanding the origin of the small angle approximation without recourse to calculus. Rather charmingly, our approach exploits the motion of the pendulum to visualise the process of taking an important limit and can be used to explore the meaning of mathematical approximation in physical systems.