Small-angle approximation

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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
\tan 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.

[edit] Geometric justification

Small angle approximation. The value of the small angle X in radians is approximately equal to its tangent.
Small angle approximation. The value of the small angle X in radians is approximately equal to its tangent.

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.

[edit] Analytic justification

The Taylor series of the trigonometric functions are

\sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
\cos\left( x \right) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
\tan\left( x \right) = 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 x2, x3, ... 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 x2 term, so their first- and second-order approximations are the same.

[edit] 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.

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