Småvinkelapproximations

Småvinkelapproximations, or the small-angle approximation, is a mathematical simplification used when angles are small. For angles θ measured in radians, sin θ ≈ θ and tan θ ≈ θ, while cos θ ≈ 1 − θ^2/2 and 1 − cos θ ≈ θ^2/2. These relations allow trigonometric functions to be treated as linear in θ in many practical problems.

The basis is the Taylor series expansions of the trigonometric functions around θ = 0: sin θ = θ − θ^3/6 + …, cos θ = 1 − θ^2/2 + θ^4/24 − …, tan θ = θ + θ^3/3 + …. When θ is small, the higher-order terms become negligible, making the functions effectively proportional to θ.

Applications of the small-angle approximation are widespread. In physics and engineering, it underpins the simple pendulum model for small displacements, yielding simple harmonic motion with a period T ≈ 2π√(L/g). In optics and computer graphics, the paraxial approximation assumes small angles relative to an optical axis to simplify ray tracing and lens equations. In astronomy and surveying, small angular separations are related to physical distances with linear approximations.

Limitations include the degradation of accuracy as θ grows; the errors are of order θ^3 for sin and tan, and θ^4 for 1 − cos. Angles must be in radians for the basic formulas to hold; using degrees requires conversion. For high-precision work, higher-order terms or exact expressions should be used.