Small-angle approximation: Difference between revisions

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[[File:Kleinwinkelnaeherungen.png|thumb|upright=1.5|Approximately equal behavior of some (trigonometric) functions for {{math|''x'' → 0}}]]

TheFor '''small- [[angle approximations''' can be used to approximate]]s, the values of the main [[trigonometric functions]] sine, providedcosine, thatand thetangent anglecan inbe questioncalculated iswith smallreasonable andaccuracy isby measuredthe infollowing [[radian]]ssimple approximations:

:<math>

\begin{align}

\sin \theta &\approx \tan \theta \approx \theta, \\[5mu]

\cos \theta &\approx 1 - \frac{tfrac12\theta^2}{2} \approx 1\\,

\tan \theta &\approx \theta

\end{align}

</math>

provided the angle is measured in [[radian]]s. Angles measured in [[degree (angle)|degrees]] must first be converted to radians by multiplying them by {{tmath|\pi/180}}.

These approximations have a wide range of uses in branches of [[physics]] and [[engineering]], including [[mechanics]], [[electromagnetics|electromagnetism]], [[optics]], [[cartography]], [[astronomy]], and [[computer science]].<ref name="Holbrow2010" /><ref name="Plesha2012"/> One reason for this is that they can greatly simplify [[differential equation]]s 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 approximation#Usage_in_science_and_engineering|order of the approximation]], <math>\textstyle \cos \theta</math> is approximated as either <math>1</math> or as <math display="inline"> 1-\frac{frac12\theta^2}{2}</math>.<ref>{{Cite web|title=Small-Angle Approximation {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/small-angle-approximation/|access-date=2020-07-22|website=brilliant.org|language=en-us}}</ref>

== 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.

<gallery widths="500px" heights="400px">

File:Small_angle_compair_odd.svg|'''Figure 1.''' A comparison of the basic [[odd function|odd]] trigonometric functions to {{mvar|θ}}. It is seen that as the angle approaches 0 the approximations become better.

File:Small_angle_compare_even.svg|'''Figure 2.''' A comparison of {{math|cos ''θ''}} to {{math|1 − {{sfrac|''θ''²|2}}}}. It is seen that as the angle approaches 0 the approximation becomes better.

</gallery>

=== Geometric ===

[[File:Small angle triangle.svg|600pxframeless|upright=2]]

The red section on the right, {{math|d}}, is the difference between the lengths of the hypotenuse, {{mvar|H}}, and the adjacent side, {{mvar|A}}. As is shown, {{mvar|H}} and {{mvar|A}} are almost the same length, meaning {{math|cos ''θ''}} is close to 1 and {{math|{{sfrac|''θ''²|2}}}} helps trim the red away.

For a small angle, {{mvar|H}} and {{mvar|A}} are almost the same length, and therefore {{math|cos ''θ''}} is nearly 1. The segment {{mvar|d}} (in red to the right) is the difference between the lengths of the hypotenuse, {{mvar|H}}, and the adjacent side, {{mvar|A}}, and has length <math>\textstyle H - \sqrt{H^2 - O^2}</math>, which for small angles is approximately equal to <math>\textstyle O^2\!/2H \approx \tfrac12 \theta^2H</math>. As a second-order approximation,

<math display="block"> \cos{\theta} \approx 1 - \frac{\theta^2}{2}.</math>

The opposite leg, {{mvar|O}}, is approximately equal to the length of the blue arc, {{mvar|s}}. GatheringThe factsarc from{{mvar|s}} geometry,has length {{math|1=''s'' = ''AθθA''}}, fromand trigonometry,by definition {{math|1=sin ''θ'' = {{sfrac|''O''|''H''}}}} and {{math|1=tan ''θ'' = {{sfrac|''O''|''A''}}}}, and fromfor thea picturesmall angle, {{math|''O'' ≈ ''s''}} and {{math|''H'' ≈ ''A''}}, which leads to:

<math display="block">\sin \theta = \frac{O}{H}\approx\frac{O}{A} = \tan \theta = \frac{O}{A} \approx \frac{s}{A} = \frac{A\theta}{A} = \theta.</math>

Or, more concisely,

Simplifying leaves,

<math display="block">\sin \theta \approx \tan \theta \approx \theta.</math>

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=== Algebraic ===

[[File:Small-angle approximation for sine function.svg|thumb|300px|The small-angle approximation for the sine function.]]

The Maclaurin[[Trigonometric expansionfunctions#Power (theseries expansion|Taylor expansionseries about 0)expansions of thetrigonometric relevantfunctions]] trigonometricsine, cosine, and tangent near functionzero isare:<ref>{{cite book| authorlink=Mary L. Boas|last=Boas| first=Mary L.|title=[[Mathematical Methods in the Physical Sciences]]|year=2006| publisher=Wiley|page=26| isbn=978-0-471-19826-0}}</ref>

<math display="block">\begin{align}

\sin \theta &= \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} \theta^{2n+1} \\

\sin \theta &= \theta - \frac{frac16\theta^3}{3!} + \fracfrac1{120}\theta^5}{5!} - \frac{\theta^7}{7!} +cdots, \cdots\[6mu]

\cos \theta &= 1 - \frac1{2}{\theta^2} + \frac1{24}\theta^4 - \cdots, \\[6mu]

<math display="block">\sintan \theta &= \theta -+ \frac{1}{3}\theta^3}{6} + \frac{\theta^52}{12015} - \frac{\theta^7}{5040}5 + \cdots </math>.

\end{align}</math>

where {{mvar|θ}} is the angle in radians. In clearer terms,

<math display="block">\sin \theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots </math>

where {{tmath|\theta}} is the angle in radians. For very small angles, higher powers of {{tmath|\theta}} become extremely small, for instance if {{tmath|1= \theta = 0.01}}, then {{tmath|1= \theta^3 = 0.000\,001}}, just one ten-thousandth of {{tmath|\theta}}. Thus for many purposes it suffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle, {{tmath|\sin\theta \approx \tan\theta \approx \theta}}, and drop the quadratic term and approximate the cosine as {{tmath|\cos\theta \approx 1}}.

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 {{val|0.000001}}, or {{sfrac|{{val|10000}}}} the first term. One can thus safely approximate:

<math display="block">\sin \theta \approx \theta</math>

If additional precision is needed the quadratic and cubic terms can also be included,

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,

<math display="block">{{tmath|\tan sin\theta \approx \sin \theta - \approx tfrac16\theta^3}},</math>

{{tmath|\cos\theta \approx 1 - \tfrac12\theta^2}}, and

{{tmath|\tan \theta &\approx \theta + \tfrac13\theta^3}}.

==== Dual numbers ====

One may also use [[dual numbers]], defined as numbers in the form <math>a + b\varepsilon</math>, with <math>a,b\in\mathbb R</math> and <math>\varepsilon</math> satisfying by definition <math>\varepsilon^2 = 0</math> and <math>\varepsilon \ne 0</math>. By using the MacLaurin series of cosine and sine, one can show that <math>\cos(\theta\varepsilon) = 1</math> and <math>\sin(\theta\varepsilon) = \theta\varepsilon</math>. Furthermore, it is not hard to prove that the [[Pythagorean identity]] holds:<math display="block">\sin^2(\theta\varepsilon) + \cos^2(\theta\varepsilon) = (\theta\varepsilon)^2 + 1^2 = \theta^2\varepsilon^2 + 1 = \theta^2 \cdot 0 + 1 = 1</math>

==Error of the approximations==

[[File:Small angle compare error.svg|thumb|upright=2300px|'''Figure 3.''' A graph of the [[relative error]]s for the small angle approximations. ({{tmath|\tan \theta \approx \theta}}, {{tmath|\sin \theta \approx \theta}}, {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }}) ]]

Near zero, the [[relative error]] of the approximations {{tmath|\cos \theta \approx 1}}, {{tmath|\sin \theta \approx \theta}}, and {{tmath|\tan \theta \approx \theta}} is quadratic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} has relative error which is quartic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error shrinks by four orders of magnitude.

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

* {{mathtmath|\cos ''θ''\theta ≈\approx 1 }} at about 0.140814 radians (8.071°)

* {{mathtmath|\tan ''θ''\theta ≈\approx ''θ''\theta}} at about 0.173017 radians (9.919°)

* {{mathtmath|\sin ''θ''\theta ≈\approx ''θ''\theta}} at about 0.244124 radians (1314.990°)

* {{mathtmath|\textstyle \cos ''θ''\theta ≈\approx 1 −- {{sfrac|''θ''^{\tfrac12\theta^2}|2}} }} at about 0.662066 radians (37.939°)

==Slide-rule approximations==

[[File:K&E Decilon slide rule left end with SRT scale.jpg | thumb | 300px | The left end of a [[Keuffel & Esser]] Deci-Lon slide rule, with a thin blue line added to show the values on the S, T, and SRT scales corresponding to sine and tangent values of 0.1 and 0.01. The S scale shows arcsine(0.1) = 5.74 degrees; the T scale shows arctangent(0.1) = 5.71 degrees; the SRT scale shows arcsine(0.01) = arctangent(0.01) = 0.01*180/pi = 0.573 degrees (to within "slide-rule accuracy").]]

[[File:K&E Decilon slide rule right end with line.jpg | thumb | 300px | The right end of a K&E Decilon slide rule with a line to show the calibration of the SRT scale at 5.73 degrees.]]

Many [[slide rule]]s – especially "trig" and higher models – include an "ST" (sines and tangents) or "SRT" (sines, radians, and tangents) scale on the front or back of the slide, for computing with sines and tangents of angles smaller than about 0.1 radian.<ref>{{cite book |title=Communications Technician M 3 & 2 |date=1965 |publisher=Bureau of Naval Personnel |page=481 |url=https://books.google.com/books?id=FYB3o7iGvb8C&pg=PA481 |access-date=7 March 2025}}</ref>

The right-hand end of the ST or SRT scale cannot be accurate to three decimal places for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees, so sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy". Some slide rules, such as the K&E Deci-Lon in the photo, calibrate 0.1 to be accurate for radian conversion, at 5.73 degrees (off by nearly 0.4% for the tangent and 0.2% for the sine for angles around 5 degrees). Others are calibrated to 5.725 degrees, to balance the sine and tangent errors at below 0.3%.

== Angle sum and difference ==

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

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=== Astronomy ===

In [[astronomy]], the [[angular size]] or angle subtended by the image of a distant object is often only a few [[arcsecond]]s (denoted by the symbol ″), so it is well suited to the small angle approximation.<ref name=Green1985/> The linear size ({{mvar|D}}) is related to the angular size ({{mvar|X}}) and the distance from the observer ({{mvar|d}}) by the simple formula:

:<math>D = X \frac{d}{206\,265{''}}</math>

where {{mvar|X}} is measured in arcseconds.

The quantity {{val|206265|u="}} is approximately equal to the number of arcseconds in 1 radian, which is the number of arcseconds in a [[circle]] ({{val|1296000|u="}}), divided by {{math|2π}}, or, the number of arcseconds in 1 radian.

The exact formula is

:<math>D = d \tan \left( X \frac{2\pi}{1\,296\,000{''}} \right)</math>

and the above approximation follows when {{math|tan ''X''}} is replaced by {{mvar|X}}.

For example, the [[parsec]] is defined by the value of d when {{mvar|D}}=1 AU, {{mvar|X}}=1 arcsecond, but the definition used is the small-angle approximation (the first equation above).

=== Motion of a pendulum ===

* [[{{main|Pendulum (mechanics)#Small-angle approximation|Small oscillations 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 mechanics|Lagrangian]] to find the indirect (energy) equation of motion.

The second-order cosine approximation is especially useful in calculating the [[potential energy]] of a [[pendulum]], which can then be applied with a [[Lagrangian mechanics|Lagrangian]] to find the indirect (energy) equation of motion. When calculating the [[frequency|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]].<ref>{{cite book |last1=Baker |first1=Gregory L. |last2=Blackburn |first2=James A. |chapter=Pendulums somewhat simple |title=The Pendulum: A Case Study in Physics |publisher=Oxford |year=2005 |doi=10.1093/oso/9780198567547.003.0002 |at=Ch. 2, {{pgs|8–26}} |isbn=0-19-856754-5 |chapter-url=https://archive.org/details/pendulumcasestud0000bake/page/8/mode/2up |chapter-url-access=limited }} {{pb}} {{cite journal |last=Bissell |first=John J. |year=2025 |title=Proof of the small angle approximation {{tmath|\sin \theta \approx \theta}} using the geometry and motion of a simple pendulum |journal=International Journal of Mathematical Education in Science and Technology |volume=56 |number=3 |pages=548–554 |doi=10.1080/0020739X.2023.2258885 |doi-access=free }}</ref>

When calculating the [[frequency|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 Interferenceinterference ===

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 {{mvar|y}} is the distance of a fringe from the center of maximum light intensity, {{mvar|m}} is the order of the fringe, {{mvar|D}} is the distance between the slits and projection screen, and {{mvar|d}} is the distance between the slits: <ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/slits.html|title=Slit Interference}}</ref><math display="block">y \approx \frac{m\lambda D}{d}

</math>

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== See also ==

* [[Skinny triangle]]

* [[Versine]]

* [[Pendulum (mechanics)#Small-angle approximation|Small oscillations of a pendulum]]

* [[Versine and haversineExsecant]]

* [[Exsecant and excosecant]]

== References ==