Small-angle approximation explained

The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:

\begin{align} \sin \theta & ≈ \theta \\ \cos \theta & ≈ 1 - (\theta ^{2})/(2) ≈ 1 \\ \tan \theta & ≈ \theta \end{align}

These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.

There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation,

style \cos \theta is approximated as either 1 or as $1-\backslash frac$.^[1]

Justifications

Graphic

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.

Geometric

The red section on the right,, is the difference between the lengths of the hypotenuse,, and the adjacent side, . As is shown, and are almost the same length, meaning is close to 1 and helps trim the red away.$$\backslash cos\; \backslash approx\; 1\; -\; \backslash frac$$

The opposite leg,, is approximately equal to the length of the blue arc, . Gathering facts from geometry,, from trigonometry, and, and from the picture, and leads to:$$\backslash sin\; \backslash theta\; =\; \backslash frac\backslash approx\backslash frac\; =\; \backslash tan\; \backslash theta\; =\; \backslash frac\; \backslash approx\; \backslash frac\; =\; \backslash frac\; =\; \backslash theta.$$

Simplifying leaves,$$\backslash sin\; \backslash theta\; \backslash approx\; \backslash tan\; \backslash theta\; \backslash approx\; \backslash theta.$$

Calculus

Using the squeeze theorem, we can prove that$$\backslash lim\_\; \backslash frac\; =\; 1,$$ which is a formal restatement of the approximation

\sin(\theta) ≈ \theta for small values of θ.

A more careful application of the squeeze theorem proves that $$\backslash lim\_\; \backslash frac\; =\; 1,$$ from which we conclude that

\tan(\theta) ≈ \theta for small values of θ.\cos 2 A \equiv 1-2 \sin ^{2} A . By letting \theta = 2 A , we get that $\backslash cos\backslash theta=1-2\backslash sin^2\backslash frac\backslash approx1-\backslash frac$.

Algebraic

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is^[2] where is the angle in radians. In clearer terms, $$\backslash sin\; \backslash theta\; =\; \backslash theta\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

It is readily seen that the second most significant (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, or the first term. One can thus safely approximate: $$\backslash sin\; \backslash theta\; \backslash approx\; \backslash theta$$

By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine, $$\backslash tan\; \backslash theta\; \backslash approx\; \backslash sin\; \backslash theta\; \backslash approx\; \backslash theta,$$

Dual numbers

One may also use dual numbers, defined as numbers in the form

a + b\varepsilon , with a,b\in R and \varepsilon satisfying by definition \varepsilon ^{2} = 0 and \varepsilon \ne 0 . By using the MacLaurin series of cosine and sine, one can show that \cos(\theta\varepsilon) = 1 and \sin(\theta\varepsilon) = \theta\varepsilon . Furthermore, it is not hard to prove that the Pythagorean identity holds:$$\backslash sin^2(\backslash theta\backslash varepsilon)\; +\; \backslash cos^2(\backslash theta\backslash varepsilon)\; =\; (\backslash theta\backslash varepsilon)^2\; +\; 1^2\; =\; \backslash theta^2\backslash varepsilon^2\; +\; 1\; =\; \backslash theta^2\; \backslash cdot\; 0\; +\; 1\; =\; 1$$

Error of the approximations

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

• at about 0.1408 radians (8.07°)
• at about 0.1730 radians (9.91°)
• at about 0.2441 radians (13.99°)
• at about 0.6620 radians (37.93°)

Angle sum and difference

The angle addition and subtraction theorems reduce to the following when one of the angles is small (β ≈ 0):

cos(α + β)	≈ cos(α) − β sin(α),
cos(α − β)	≈ cos(α) + β sin(α),
sin(α + β)	≈ sin(α) + β cos(α),
sin(α − β)	≈ sin(α) − β cos(α).

Specific uses

Astronomy

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

D = X (d)/(206265 {'') }

where is measured in arcseconds.

The quantity is approximately equal to the number of arcseconds in a circle, divided by, or, the number of arcseconds in 1 radian.

The exact formula is

D = d \tan \left( X (2 \pi)/(1296000 {'') } \right)

and the above approximation follows when is replaced by .

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.

Optics

In optics, the small-angle approximations form the basis of the paraxial approximation.

Wave Interference

The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where is the distance of a fringe from the center of maximum light intensity, is the order of the fringe, is the distance between the slits and projection screen, and is the distance between the slits: ^[3] $$y\; \backslash approx\; \backslash frac$$

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 behavior.

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.

Interpolation

The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values:

Example: sin(0.755)where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.

See also

• Skinny triangle
• Small oscillations of a pendulum
• Versine and haversine
• Exsecant and excosecant

Notes and References

1. Web site: Small-Angle Approximation Brilliant Math & Science Wiki. 2020-07-22. brilliant.org. en-us.
2. Book: Boas, Mary L.. Mary L. Boas

   . Mary L. Boas. Mathematical Methods in the Physical Sciences. 2006. Wiley. 26. 978-0-471-19826-0.

3. Web site: Slit Interference.