Small-angle approximation

Small-angle approximation

Small-angle approximation is a useful simplification of the laws of trigonometry which is only approximately true for finite angles, but correct in the limit as the angle approaches zero. It involves linearization of the trigonometric functions (truncation of their Taylor series) so that, when the angle "x" is measured in radians,

:sin x simeq x

:cos x simeq 1 or cos x simeq 1 - frac{x^2}{2} for the second-order approximation

: an x simeq x

Small angle approximation is useful in many areas of physical science, including optics (where it forms the basis of the paraxial approximation), cartography, and astronomy.

Geometric justification

When one angle of a right triangle is small, its hypotenuse is approximately equal in length to the leg adjacent to the small angle, so the cosine is approximately 1. The short leg is approximately equal to the arc from the long leg to the hypotenuse, so the sine and tangent are both approximated by the value of the angle in radians.

Analytic justification

The Taylor series of the trigonometric functions are

:sinleft( x ight) = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots :cosleft( x ight) = 1 - frac{x^2}{2!} + frac{x^4}{4!} - frac{x^6}{6!} + cdots : anleft( x ight) = x + frac{x^3}{3} + frac{2 x^5}{15} + frac{17 x^7}{315} + cdots

When the angle "x" is less than one radian, its powers "x"2, "x"3, ... decrease rapidly, so only a few are needed. The highest power included is called the order of the approximation. Neither sin("x") nor tan("x") has an "x"2 term, so their first- and second-order approximations are the same.

Specific uses

In astronomy, the angle subtended by the image of a distant object is often only a few arcseconds, so it is well suited to the small angle approximation. The linear size ("D") is related to the angular size ("X") and the distance from the observer ("d") by the simple formula

:"D" = "X" · "d" / 206,265

where "X" is measured in arcseconds.

The number 206,265 is approximately equal to the number of arcseconds in a circle (1,296,000), divided by 2π.

The exact formula is

:"D" = 2 "d" tan("X"·π/1,296,000)

and the above approximation follows when tan("X") is replaced by "X".

The second order Cos approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion.

The small angle approximation also appears in structural mechanics especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling) leading to significant simplifications, though at a cost in accuracy and insight into the true behaviour.

The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Small-angle X-ray scattering — (SAXS) is a small angle scattering (SAS) technique where the elastic scattering of X rays (wavelength 0.1 ... 0.2 nm) by a sample which has inhomogeneities in the nm range, is recorded at very low angles (typically 0.1 10°). This angular range… …   Wikipedia

  • Biological small-angle scattering — Small angle scattering is a fundamental method for structure analysis of materials, including biological materials. Small angle scattering allows one to study the structure of a variety of objects such as solutions of biological macromolecules,… …   Wikipedia

  • Grazing-incidence small-angle X-ray scattering — (GISAXS) is a scattering technique most commonly done at synchrotron radiation facilities. A related technique also exists for neutron scattering (GISANS). 300px|thumb|right|Scheme 1: GISAXS scattering geometry. The incident beam strikes the… …   Wikipedia

  • Angle of repose — For the Wallace Stegner novel, see Angle of Repose (novel). For the friction angle between two solid objects, see Friction. For the Sleepytime Gorilla Museum song of the same name, see In Glorious Times. Angle of repose The angle of repose or,… …   Wikipedia

  • Born approximation — Not to be confused with the Born–Oppenheimer approximation. In scattering theory and, in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point… …   Wikipedia

  • Paraxial approximation — In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens).cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics |… …   Wikipedia

  • Pendulum (mathematics) — The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small angle oscillations. Simple gravity… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Optics — For the book by Sir Isaac Newton, see Opticks. Optical redirects here. For the musical artist, see Optical (artist). Optics includes study of dispersion of light. Optics is the branch of …   Wikipedia

  • Foucault pendulum — The Foucault pendulum (pronEng|fuːˈkoʊ foo KOH ), or Foucault s pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth.The experimentThe experimental apparatus consists of …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.