The small-angle approximation linearizes trigonometric functions near zero. When an angle θ is sufficiently small (in radians), you can replace sinθ with θ, turning many nonlinear equations into linear ones and producing simple harmonic motion.
Taylor series near θ = 0
sin θ = θ - θ^3/6 + O(θ^5)
cos θ = 1 - θ^2/2 + O(θ^4)
So the leading approximation is
sin θ ≈ θ
and the error starts at order θ^3.
Radians matter
These series are valid when θ is measured in radians. Using degrees directly breaks the approximation numerically.
Pendulum linearization
The exact (ideal) pendulum equation is nonlinear:
θ¨ + (g/L) sin θ = 0
With sinθ ≈ θ, it becomes
θ¨ + (g/L) θ = 0
ω = sqrt(g/L)
T0 = 2π sqrt(L/g)
This predicts a period independent of amplitude, but only within the small-angle regime.
Beyond small angles
For larger amplitudes, the period increases with amplitude. A common series correction for amplitude θ0 is:
T ≈ T0 (1 + θ0^2/16 + 11 θ0^4/3072 + ...)
The exact expression involves an elliptic integral.
Common pitfalls
- The approximation is about small θ, not small sinθ in degrees.
- It is a local linearization; errors can accumulate in long-time predictions.