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== Justifications ==
=== Geometric ===
[[File:Small angle triangle.svg|
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}}.
<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,
<math display="block">\sin \theta \approx \tan \theta \approx \theta.</math>
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{{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|
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.
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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°)
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[[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 trigonometric functions of angles smaller than about 0.1 radian (or within about 0.1 radian of <math>\pi/2</math>, that is, within 5.7 degrees of 90 degrees, in the case of cosines and cotangents).<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#v=onepage&q&f=false |access-date=7 March 2025}}</ref>▼
▲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
The right-hand end of the ST or SRT scale cannot be accurate for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees. Some slide rules, such as the K&E Deci-Lon in the photo, calibrate 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%. Either way, sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy".▼
▲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 ==
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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="}})
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}}
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===
In optics, the small-angle approximations form the basis of the [[paraxial approximation]].
=== Wave
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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