Below roughly what swing angle does a simple pendulum behave as a simple harmonic oscillator?

1. The restoring force on a pendulum bob is -m g sin(theta), which is not proportional to the angle in general. 2. For small angles, sin(theta) is very close to theta itself, so the force becomes -m g theta. 3. That is Hooke's law in the form F = -k x, so the motion is simple harmonic. 4. The approximation holds for angles below about 15 degrees, where sin(theta) and theta differ by about 1 percent or less. 5. At 45 or 90 degrees the approximation collapses and the motion is no longer simple harmonic. _Source: OpenStax College Physics (CC BY 4.0), Ch 16 "Oscillatory Motion and Waves", section 16.4 The Simple Pendulum_