Small-angle approximation: Difference between revisions

Revision as of 07:18, 13 April 2025

Latest revision as of 03:15, 18 August 2025

(4 intermediate revisions by 2 users not shown)

Line 61:

{{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==

Line 82 ⟶ 79:

[[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#v=onepage&q&f=false |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 ==

Line 102 ⟶ 99:

=== 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}}.

Line 124 ⟶ 117:

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>