Small-angle approximation

Approximately equal behavior of some (trigonometric) functions for x → 0

The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the basic trigonometric functions to a second-order approximation. This truncation gives:

\sin \theta \approx \theta

\cos \theta \approx 1 - \frac{\theta^2}{2}

\tan \theta \approx \theta

where θ is the angle in radians.

The small angle approximation is useful in many areas of physics, including mechanics, electromagnetics, optics (where it forms the basis of the paraxial approximation), cartography, astronomy, and so on.

Justifications

Graphic

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes.

Figure 1. A comparison of the basic odd trigonometric functions to θ. It is seen that as the angle approaches 0 the approximations become better.
Figure 2. A comparison of cos(θ) to 1 - θ²/2. It is seen that as the angle approaches 0 the approximation becomes better.

Geometric

The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the same length, meaning cos θ is close to 1 and $\theta^2/2$ helps trim the red away.

\cos{\theta} \approx 1 - \frac{\theta^2}{2}

The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = A*θ, from trigonometry, sin θ = O/H and tan θ = O/A, and from the picture, $O \approx s$ and $H \approx A$ leads to:

\sin \theta = {O\over H}\approx{O\over A} = \tan \theta = {O\over A} \approx {s\over A} = {{A*\theta}\over A} = \theta

Simplifying leaves,

\sin \theta \approx \tan \theta \approx \theta

Algebraic

The small-angle approximation for the sine function.

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is^[1]

\sin \theta =\sum _{{n=0}}^{{\infty }}{\frac {(-1)^{n}}{(2n+1)!}}\theta ^{{2n+1}}=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots

where θ is the angle in radians. In clearer terms,

\sin \theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots

It is readily seen that the second most significant (viz., the third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of 0.000001, or one-ten-thousandth the first term. One can thus safely approximate:

\sin \theta \approx \theta

By extension, since the cosine of a small angle is very, very nearly one, and the tangent is given by the sine divided by the cosine,

\tan \theta \approx \sin \theta \approx \theta

Error of the approximations

Figure 3. A graph of the relative errors for the small angle approximations.

Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:

tan θ ≈ θ at about 0.176 radians(10°).
sin θ ≈ θ at about 0.244 radians (14°).
cos θ ≈ 1 - θ²/2 at about 0.664 radians (38°).

Specific uses

Astronomy

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 = d tan(X·2π/1,296,000)

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

Motion of a pendulum

The second-order cosine 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.

When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.

Structural mechanics

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

Piloting

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.

References

↑ Boas, Mary L. (2006). Mathematical Methods in the Physical Sciences. p. 26: Wiley. p. 839. ISBN 978-0-471-19826-0.

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