Small angle approximation

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Almost the same behavior of some (trigonometric) functions for x → 0

The small-angle approximation is understood as the mathematical approximation in which it is assumed that the angle is sufficiently small that its sine or tangent can be replaced by the angle itself (in radians ) and the cosine by :

Derivation

The basis of this approach is the respective Maclaurin series of the angle function (see also Taylor series ):

For one can neglect the summands with a higher power of compared to the preceding terms, so that the above Approximate results.

To assess the angle up to which the approximations are permissible within the framework of accepted error limits , some relative deviations are specified:

Relative deviation at
Approximation
instead of
instead of
instead of

Applications

The small-angle approximation is particularly important in physics , where many problems can be solved analytically with the help of the small-angle approximation , which would otherwise lead to complicated elliptic integrals if the angle functions were included . Application examples of the small angle approximation are the mathematical pendulum , the evaluation of the diffraction at the slit , the paraxial optics as well as the approximation of parabola and circular arc in the treatment of lenses and concave mirrors near the optical axis .

Moderate angle changes> 7 °

In technical mechanics, it is also common to consider moderate changes in angle. In order to avoid that the cosine falls out completely in the small angle approximation, the second term of the Taylor series expansion is also taken into account, so that:

.

An application example is the theory of slightly curved shell structures: Since the curvature has a decisive influence on the load-bearing behavior, it must be taken into account; at the same time, the approximation should reduce the computational effort.

The more precise approximation now results in the following properties:

Relative deviation at
Approximation
instead of

literature

  • Berthold Schuppar: Elementary Numerical Mathematics . Springer, 2013, ISBN 978-3-322-80307-8 , pp. 67-70 .