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Small-angle approximation

## Summary

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:

{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\cos \theta &\approx 1-{\frac {\theta ^{2}}{2}}\approx 1\\\tan \theta &\approx \theta \end{aligned}}}

These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science.[1][2] 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, ${\displaystyle \textstyle \cos \theta }$ is approximated as either ${\displaystyle 1}$ or as ${\textstyle 1-{\frac {\theta ^{2}}{2}}}$.[3]

## 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, 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 θ2/2 helps trim the red away.

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

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

${\displaystyle \sin \theta ={\frac {O}{H}}\approx {\frac {O}{A}}=\tan \theta ={\frac {O}{A}}\approx {\frac {s}{A}}={\frac {A\theta }{A}}=\theta .}$

Simplifying leaves,

${\displaystyle \sin \theta \approx \tan \theta \approx \theta .}$

### Calculus

Using the squeeze theorem,[4] we can prove that

${\displaystyle \lim _{\theta \to 0}{\frac {\sin(\theta )}{\theta }}=1,}$

which is a formal restatement of the approximation ${\displaystyle \sin(\theta )\approx \theta }$  for small values of θ.

A more careful application of the squeeze theorem proves that

${\displaystyle \lim _{\theta \to 0}{\frac {\tan(\theta )}{\theta }}=1,}$

from which we conclude that ${\displaystyle \tan(\theta )\approx \theta }$  for small values of θ.

Finally, L'Hôpital's rule tells us that

${\displaystyle \lim _{\theta \to 0}{\frac {\cos(\theta )-1}{\theta ^{2}}}=\lim _{\theta \to 0}{\frac {-\sin(\theta )}{2\theta }}=-{\frac {1}{2}},}$

which rearranges to ${\textstyle \cos(\theta )\approx 1-{\frac {\theta ^{2}}{2}}}$  for small values of θ. Alternatively, we can use the double angle formula ${\displaystyle \cos 2A\equiv 1-2\sin ^{2}A}$ . By letting ${\displaystyle \theta =2A}$ , we get that ${\textstyle \cos \theta =1-2\sin ^{2}{\frac {\theta }{2}}\approx 1-{\frac {\theta ^{2}}{2}}}$ .

### Algebraic

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is[5]

{\displaystyle {\begin{aligned}\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 \end{aligned}}}

where θ is the angle in radians. In clearer terms,
${\displaystyle \sin \theta =\theta -{\frac {\theta ^{3}}{6}}+{\frac {\theta ^{5}}{120}}-{\frac {\theta ^{7}}{5040}}+\cdots }$

It is readily seen that the second most significant (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 1/10000 the first term. One can thus safely approximate:

${\displaystyle \sin \theta \approx \theta }$

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,

${\displaystyle \tan \theta \approx \sin \theta \approx \theta ,}$

#### Dual numbers

One may use dual numbers, denoted ε, often considered similar to an infinitesimal amount, with a square of 0. By using the Maclaurin series of cosine and sine and substituting θ=θε, the result is cos(θε)=1 and sin(θε)=θε. These approximations satisfy the Pythagorean identity:

cos²(θε)+sin²(θε)=1²+(θε)²=1+θ²ε²=1+θ²0=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:

• cos θ ≈ 1 − θ2/2 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.[6] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:

${\displaystyle D=X{\frac {d}{206\,265{''}}}}$

where X is measured in arcseconds.

The quantity 206265 is approximately equal to the number of arcseconds in a circle (1296000), divided by , or, the number of arcseconds in 1 radian.

The exact formula is

${\displaystyle D=d\tan \left(X{\frac {2\pi }{1\,296\,000{''}}}\right)}$

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.

### 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 y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen, and d is the distance between the slits: [7]

${\displaystyle y\approx {\frac {m\lambda D}{d}}}$

### 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)

{\displaystyle {\begin{aligned}\sin(0.755)&=\sin(0.75+0.005)\\&\approx \sin(0.75)+(0.005)\cos(0.75)\\&\approx (0.6816)+(0.005)(0.7317)\\&\approx 0.6853.\end{aligned}}}

where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table.