Equations · equation
Small-Angle Approximation
Calculus
- Equation
- sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2 (θ in radians, θ ≪ 1)
- Field
- Calculus
- Symbols
- θ an angle in radians, small compared with 1
Significance
The first terms of three Taylor series, and the reason the simple pendulum has a simple period, a lens has a focal length, and a diffraction pattern has evenly spaced fringes. At 10° the error in sin θ ≈ θ is half a percent; at 30° it is 5 %. Every one of those results is exact only in the limit, and the honest ones say so.
Open on the Equations shelf — the interactive card, the filters, and everything alongside it.