Small-angle approximation: Difference between revisions

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{{Short description|Simplification of the basic trigonometric functions}}

{{Unreferenced|date=December 2009}}

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

'''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 (mathematics)|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 [[radian]]s,

For small [[angle]]s, the [[trigonometric functions]] sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:

:<math>\sin x \simeq x</math>

:<math>

:<math>\cos x \simeq 1</math> or <math>\cos x \simeq 1 - \frac{x^2}{2}</math> for the [[orders of approximation|second-order approximation]]

\begin{align}

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

\cos \theta &\approx 1 - \tfrac12\theta^2 \approx 1,

\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}}.

:<math>\tan x \simeq x</math>

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.

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

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-\frac12\theta^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>

==Geometric justification==

[[Image:Small-angle.png|frame|center|Small angle approximation. The value of the small angle ''X'' in radians is approximately equal to its tangent.]]

== Justifications ==

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.

=== Geometric ===

==Analytic justification==

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

The Taylor series of the trigonometric functions are

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>\sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots </math>

:<math display="block"> \cos{\left( xtheta} \right) =approx 1 - \frac{x\theta^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots .</math>

:<math>\tan\left( x \right) = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \frac{17 x^7}{315} + \cdots</math>

The opposite leg, {{mvar|O}}, is approximately equal to the length of the blue arc, {{mvar|s}}. The arc {{mvar|s}} has length {{math|1=''θA''}}, and by definition {{math|1=sin ''θ'' = {{sfrac|''O''|''H''}}}} and {{math|1=tan ''θ'' = {{sfrac|''O''|''A''}}}}, and for a small angle, {{math|''O'' ≈ ''s''}} and {{math|''H'' ≈ ''A''}}, which leads to:

When the angle ''x'' is less than one radian, its powers ''x''², ''x''³, ... 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 ''x''² term, so their first- and second-order approximations are the same.

<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,

==Specific uses==

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

In astronomy, the angle subtended by the image of a distant object is often only a few [[arcsecond]]s, 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

=== Calculus ===

:''D'' = ''X'' · ''d'' / 206,265

Using the [[squeeze theorem]],<ref name=Larson2006/> we can prove that

<math display="block">

\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1,

</math> which is a formal restatement of the approximation <math>\sin(\theta) \approx \theta</math> for small values of ''θ''.

A more careful application of the squeeze theorem proves that <math display="block">

where ''X'' is measured in arcseconds.

\lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1,

</math> from which we conclude that <math>\tan(\theta) \approx \theta</math> for small values of ''θ''.

Finally, [[L'Hôpital's rule]] tells us that <math display="block">

The number 206,265 is approximately equal to the number of arcseconds in a [[circle]] (1,296,000), divided by 2π.

\lim_{\theta \to 0} \frac{\cos(\theta)-1}{\theta^2} = \lim_{\theta \to 0} \frac{-\sin(\theta)}{2\theta} = -\frac{1}{2},

</math> which rearranges to <math display="inline">\cos(\theta) \approx 1 - \frac{\theta^2}{2}</math> for small values of ''θ''. Alternatively, we can use the [[List of trigonometric identities#Double-angle formulae|double angle formula]] <math>\cos 2A \equiv 1-2\sin^2 A</math>. By letting <math>\theta = 2A</math>, we get that <math display="inline">\cos\theta=1-2\sin^2\frac{\theta}{2}\approx1-\frac{\theta^2}{2}</math>.

=== Algebraic ===

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

The [[Trigonometric functions#Power series expansion|Taylor series expansions of trigonometric functions]] sine, cosine, and tangent near zero are:<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 &= \theta - \frac16\theta^3 + \frac1{120}\theta^5 - \cdots, \\[6mu]

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

\tan \theta &= \theta + \frac{1}{3}\theta^3 + \frac{2}{15}\theta^5 + \cdots.

\end{align}</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}}.

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

{{tmath|\sin\theta \approx \theta - \tfrac16\theta^3}},

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

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

==Error of the approximations==

[[File:Small angle compare error.svg|thumb|300px|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:

* {{tmath|\cos \theta \approx 1 }} at about 0.14 radians (8.1°)

* {{tmath|\tan \theta \approx \theta}} at about 0.17 radians (9.9°)

* {{tmath|\sin \theta \approx \theta}} at about 0.24 radians (14.0°)

* {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} at about 0.66 radians (37.9°)

==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):

:{|

|style="text-align:right;"| cos(''α'' + ''β'') ||≈ cos(''α'') − ''β'' sin(''α''),

|-

|style="text-align:right;"| cos(''α'' − ''β'') ||≈ cos(''α'') + ''β'' sin(''α''),

|-

|style="text-align:right;"| sin(''α'' + ''β'') ||≈ sin(''α'') + ''β'' cos(''α''),

|-

|style="text-align:right;"| 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 [[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}}.

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).

:''D'' = 2 ''d'' tan(''X''·π/1,296,000)

=== Motion of a pendulum ===

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

{{main|Pendulum (mechanics)#Small-angle approximation}}

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>

===Optics===

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.

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

=== Wave interference ===

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 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>

=== 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 ===

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]].

The formulas for [[#Angle sum and difference|addition and subtraction involving a small angle]] may be used for [[interpolating]] between [[trigonometric table]] values:

Example: sin(0.755)

{{DEFAULTSORT:Small-Angle Approximation}}

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

[[Category:Geometry]]

\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{align}</math>

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 ==

[[de:Kleinwinkelnäherung]]

* [[Skinny triangle]]

[[id:Pendekatan paraksial]]

* [[Versine]]

[[he:קירוב זווית קטנה]]

* [[Exsecant]]

== References ==

{{reflist|refs=

<ref name=Holbrow2010>{{citation

| title=Modern Introductory Physics | postscript=.

| first1=Charles H. | last1=Holbrow | first2=James N. | last2=Lloyd

| first3=Joseph C. | last3=Amato | first4=Enrique | last4=Galvez

| first5=M. Elizabeth | last5=Parks | display-authors=1

| edition=2nd | publisher=Springer Science & Business Media

| year=2010 | isbn=978-0387790794 | pages=30–32

| url=https://books.google.com/books?id=KLT_FyQyimUC&pg=PA30 }}</ref>

<ref name=Plesha2012>{{citation

| title=Engineering Mechanics: Statics and Dynamics

| first1=Michael | last1=Plesha | first2=Gary | last2=Gray

| first3=Francesco | last3=Costanzo | display-authors=1

| edition=2nd | publisher=McGraw-Hill Higher Education

| year=2012 | isbn=978-0077570613 | page=12 | postscript=.

| url=https://books.google.com/books?id=xWt6CgAAQBAJ&pg=PA12 }}</ref>

<ref name=Larson2006>{{citation

| title=Calculus of a Single Variable: Early Transcendental Functions

| first1=Ron | last1=Larson | first2=Robert P. | last2=Hostetler

| first3=Bruce H. | last3=Edwards | display-authors=1

| edition=4th | publisher=Cengage Learning

| year=2006 | isbn=0618606254 | page=85 | postscript=.

| url=https://books.google.com/books?id=E5V4vTqAgAIC&pg=PA85 }}</ref>

<ref name=Green1985>{{citation

| title=Spherical Astronomy | postscript=.

| first1=Robin M. | last1=Green

| publisher=Cambridge University Press

| year=1985 | isbn=0521317797 | page=19

| url=https://books.google.com/books?id=wOpaUFQFwTwC&pg=PA19 }}</ref>

}}

{{DEFAULTSORT:Small-Angle Approximation}}

[[Category:Trigonometry]]

[[Category:Equations of astronomy]]