Small-angle approximation

Under the small angle approximation, the mathematical approximation is understood, it is assumed when the angle x is sufficiently small so that it can be replaced by its sine or tangent of 1 by the angle itself ( in radians) and the cosine.

Following this approach, the respective Maclaurin series, the angle function (see Taylor series ):

For one can neglect the higher potency summand of x over the previous links, so that there are the approximations:

To judge up to what angle the approximations within accepted limits of error are permitted some relative deviations are given:

Importantly, the small angle approximation is particularly in physics, where many problems can be solved exactly by means of the small angle approximation analytically, which would otherwise result in the involvement of the trigonometric functions to complex elliptic integrals. Examples of application of the small angle approximation is the mathematical pendulum, the evaluation of the diffraction at the nip, the paraxial optics, and the approximation of the parabola and a circular arc in the treatment of lenses and concave mirrors in the vicinity of the optical axis.

• Geometry
• Numerical Mathematics