Editing Radian (section)

== Definition ==

One radian is defined as the angle at the center of a circle in a plane that [[wikt:subtend|subtends]] an arc whose length equals the radius of the circle.<ref>{{ citation | last1 = Protter | first1 = Murray H. | last2 = Morrey | first2 = Charles B. Jr. | title = College Calculus with Analytic Geometry | edition = 2nd | location = Reading | publisher = [[Addison-Wesley]] | year = 1970 | page = APP-4 | lccn = 76087042 }}</ref> More generally, the [[magnitude (mathematics)|magnitude]] in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, <math>\theta = \frac{s}{r}</math>, where {{mvar|θ}} is the magnitude in radians of the subtended angle, {{mvar|s}} is arc length, and {{mvar|r}} is radius. A [[right angle]] is exactly <math>\frac{\pi}{2}</math> radians.{{sfn|International Bureau of Weights and Measures|2019|p=151}}

One [[complete revolution]], expressed as an angle in radians, is the length of the circumference divided by the radius, which is <math>\frac {2\pi r}{r}</math>, or {{math|2''π''}}. Thus, {{math|2''π''}}&nbsp;radians is equal to 360&nbsp;degrees. The relation {{math|1=2''π'' rad = 360°}} can be derived using the formula for [[arc length]], <math display="inline">\ell_{\text{arc}}=2\pi r\left(\tfrac{\theta}{360^{\circ}}\right)</math>. Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle, <math display="inline">1=2\pi\left(\tfrac{1\text{ rad}}{360^{\circ}}\right)</math>. This can be further simplified to <math display="inline">1=\tfrac{2\pi\text{ rad}}{360^{\circ}}</math>. Multiplying both sides by {{math|360°}} gives {{math|360° {{=}} 2''π'' rad}}.

=== Unit symbol ===
The [[International Bureau of Weights and Measures]]{{sfn|International Bureau of Weights and Measures|2019|p=151}} and [[International Organization for Standardization]]<ref>{{cite web |title=ISO 80000-3:2006 Quantities and Units - Space and Time |date=17 January 2017 |url=https://www.iso.org/standard/31888.html}}</ref> specify '''rad''' as the symbol for the radian. Alternative symbols that were in use in 1909 are <sup>c</sup> (the superscript letter c, for "circular measure"), the letter r, or a superscript {{sup|R}},<ref name="Hall_1909" /> but these variants are infrequently used, as they may be mistaken for a [[degree symbol]] (°) or a radius (r). Hence an angle of 1.2 radians would be written today as 1.2&nbsp;rad; archaic notations include 1.2&nbsp;r, 1.2{{sup|rad}}, 1.2{{sup|c}}, or 1.2{{sup|R}}.

In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the [[degree sign]] {{char|°}} is used.

=== Dimensional analysis ===
{{see also|#As an SI unit}}

Plane angle may be defined as {{math|1=''[[θ]]'' = ''s''/''r''}}, where {{mvar|θ}} is the magnitude in radians of the subtended angle, {{mvar|s}} is circular arc length, and {{mvar|r}} is radius. One radian corresponds to the angle for which {{math|1=''s'' = ''r''}}, hence {{math|1=1 radian = 1&nbsp;m/m}} = 1.<ref>{{harvnb|International Bureau of Weights and Measures|2019|p=151}}: "One radian corresponds to the angle for which {{math|1=''s'' = ''r''}}"</ref> However, {{math|rad}} is only to be used to express angles, not to express ratios of lengths in general.{{sfn|International Bureau of Weights and Measures|2019|p=151}} A similar calculation using [[Circular sector#Area|the area of a circular sector]] {{math|1=''θ'' = 2''A''/''r''<sup>2</sup>}} gives 1 radian as 1&nbsp;m<sup>2</sup>/m<sup>2</sup> = 1.<ref>{{harvnb|Quincey|2016|p=844}}: "Also, as alluded to in {{harvnb|Mohr|Phillips|2015}}, the radian can be defined in terms of the area ''A'' of a sector ({{math|1=''A'' = {{sfrac|1|2}} ''θ'' ''r''<sup>2</sup>}}), in which case it has the units m<sup>2</sup>⋅m<sup>−2</sup>."</ref> The key fact is that the radian is a [[dimensionless unit]] equal to [[1]]. In SI 2019, the SI radian is defined accordingly as {{nowrap|1=1 rad = 1}}.<ref>{{harvnb|International Bureau of Weights and Measures|2019|p=151}}: "One radian corresponds to the angle for which {{math|1=''s'' = ''r''}}, thus {{math|1=1 rad = 1}}."</ref> It is a long-established practice in mathematics and across all areas of science to make use of {{math|1=rad = 1}}.{{sfn|International Bureau of Weights and Measures|2019|p=137}}<ref>{{cite book |last1=Bridgman |first1=Percy Williams |url=https://archive.org/details/dimensionalanaly00bridrich/page/n13/mode/2up |title=Dimensional analysis |date=1922 |location=New Haven |publisher=Yale University Press |quote=Angular amplitude of swing [...] No dimensions.}}</ref>

Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations".<ref>{{cite journal |last1=Prando |first1=Giacomo |date=August 2020 |title=A spectral unit |journal=Nature Physics |volume=16 |issue=8 |pages=888 |bibcode=2020NatPh..16..888P |doi=10.1038/s41567-020-0997-3 |s2cid=225445454|doi-access=free }}</ref> For example, an object hanging by a string from a pulley will rise or drop by {{math|1=''y'' = ''rθ''}} centimetres, where {{mvar|r}} is the magnitude of the radius of the pulley in centimetres and {{mvar|θ}} is the magnitude of the angle through which the pulley turns in radians. When multiplying {{mvar|r}} by {{mvar|θ}}, the unit radian does not appear in the product, nor does the unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the [[angular velocity]] of a rolling wheel, {{math|1=''ω'' = ''v''/''r''}}, radians appear in the units of {{mvar|ω}} but not on the right hand side.<ref>{{cite book |last1=Leonard |first1=William J. |url=https://books.google.com/books?id=ShcCF5Gb408C&pg=PA262 |title=Minds-on Physics: Advanced topics in mechanics |date=1999 |publisher=Kendall Hunt |isbn=978-0-7872-5412-4 |page=262 |language=en}}</ref> Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".<ref>{{cite journal |last1=French |first1=Anthony P. |date=May 1992 |title=What happens to the 'radians'? (comment) |journal=The Physics Teacher |volume=30 |issue=5 |pages=260–261 |doi=10.1119/1.2343535}}</ref> Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".<ref>{{cite journal |last1=Oberhofer |first1=E. S. |date=March 1992 |title=What happens to the 'radians'? |journal=The Physics Teacher |volume=30 |issue=3 |pages=170–171 |bibcode=1992PhTea..30..170O |doi=10.1119/1.2343500}}</ref>

In 1993 the [[American Association of Physics Teachers]] Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of [[angle measure]] (rad), [[angular speed]] (rad/s), [[angular acceleration]] (rad/s<sup>2</sup>), and [[torsion constant#Torsional Rigidity (GJ) and Stiffness (GJ/L)|torsional stiffness]] (N⋅m/rad), and not in the quantities of [[torque]] (N⋅m) and [[angular momentum]] (kg⋅m<sup>2</sup>/s).<ref>{{cite journal |last1=Aubrecht |first1=Gordon J. |last2=French |first2=Anthony P. |last3=Iona |first3=Mario |last4=Welch |first4=Daniel W. |date=February 1993 |title=The radian—That troublesome unit |journal=The Physics Teacher |volume=31 |issue=2 |pages=84–87 |bibcode=1993PhTea..31...84A |doi=10.1119/1.2343667}}</ref>

At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a [[base unit of measurement]] for a [[base quantity]] (and dimension) of "plane angle".<ref>{{harvnb|Brinsmade|1936}}; {{harvnb|Romain|1962}}; {{harvnb|Eder|1982}}; {{harvnb|Torrens|1986}}; {{harvnb|Brownstein|1997}}; {{harvnb|Lévy-Leblond|1998}}; {{harvnb|Foster|2010}}; {{harvnb|Mills|2016}}; {{harvnb|Quincey|2021}}; {{harvnb|Leonard|2021}}; {{harvnb|Mohr|Shirley|Phillips|Trott|2022}}</ref>{{sfn|Mohr|Phillips|2015}}<ref name="Quincey">{{cite journal |last1=Quincey |first1=Paul |last2=Brown |first2=Richard J C |date=1 June 2016 |title=Implications of adopting plane angle as a base quantity in the SI |journal=Metrologia |volume=53 |issue=3 |pages=998–1002 |arxiv=1604.02373 |bibcode=2016Metro..53..998Q |doi=10.1088/0026-1394/53/3/998 |s2cid=119294905}}</ref> Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the [[area of a circle]], {{math|1=π''r''<sup>2</sup>}}. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".{{sfn|Quincey|2016}} A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.<ref name="Quincey" />

In particular, Quincey identifies Torrens' proposal to introduce a constant {{mvar|[[η]]}} equal to 1 inverse radian (1&nbsp;rad<sup>−1</sup>) in a fashion similar to the [[Vacuum permittivity#Historical origin of the parameter ε0|introduction of the constant ''ε''<sub>0</sub>]].{{sfn|Quincey|2016}}{{efn|Other proposals include the abbreviation "rad" {{harv|Brinsmade|1936}}, the notation <math>\langle \theta \rangle</math> {{harv|Romain|1962}}, and the constants [[ם]] {{harv|Brownstein|1997}}, ◁ {{harv|Lévy-Leblond|1998}}, ''k'' {{harv|Foster|2010}}, ''θ''<sub>C</sub> {{harv|Quincey|2021}}, and <math>{\cal C} = \frac{2\pi}{\Theta}</math> {{harv|Mohr|Shirley|Phillips|Trott|2022}}.}} With this change the formula for the angle subtended at the center of a circle, {{math|1=''s'' = ''rθ''}}, is modified to become {{math|1=''s'' = ''ηrθ''}}, and the [[Taylor series]] for the [[sine]] of an angle {{mvar|θ}} becomes:<ref name="Quincey" />{{sfn|Torrens|1986}}
<math display="block">\operatorname{Sin} \theta = \sin \ x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \eta \theta - \frac{(\eta \theta)^3}{3!} + \frac{(\eta \theta)^5}{5!} - \frac{(\eta \theta)^7}{7!} + \cdots ,</math>
where <math>x = \eta \theta = \theta/\text{rad}</math> is the angle in radians.
The capitalized function {{math|Sin}} is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed,{{sfn|Torrens|1986}} while {{math|sin}} is the traditional function on [[pure number]]s which assumes its argument is a dimensionless number in radians.{{sfn|Mohr|Shirley|Phillips|Trott|2022|p=6}} The capitalised symbol <math>\operatorname{Sin}</math> can be denoted <math>\sin</math> if it is clear that the complete form is meant.<ref name="Quincey" />{{sfn|Mohr|Shirley|Phillips|Trott|2022|pp=8-9}}

Current SI can be considered relative to this framework as a [[natural unit]] system where the equation {{math|1=''η'' = 1}} is assumed to hold, or similarly, {{nowrap|1=1 rad = 1}}. This ''radian convention'' allows the omission of {{mvar|η}} in mathematical formulas.{{sfn|Quincey|2021}}

Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.<ref>{{cite journal |last1=Quincey |first1=Paul |last2=Brown |first2=Richard J C |date=1 August 2017 |title=A clearer approach for defining unit systems |journal=Metrologia |volume=54 |issue=4 |pages=454–460 |arxiv=1705.03765 |bibcode=2017Metro..54..454Q |doi=10.1088/1681-7575/aa7160 |s2cid=119418270}}</ref> For example, the [[Boost (C++ libraries)|Boost]] units library defines angle units with a <code>plane_angle</code> dimension,<ref>{{cite web |last1=Schabel |first1=Matthias C. |last2=Watanabe |first2=Steven |title=Boost.Units FAQ – 1.79.0 |url=https://www.boost.org/doc/libs/1_79_0/doc/html/boost_units/FAQ.html#boost_units.FAQ.Angle_Are_Units |access-date=5 May 2022 |website=www.boost.org |quote=Angles are treated as units}}</ref> and [[Mathematica]]'s unit system similarly considers angles to have an angle dimension.{{sfn|Mohr|Shirley|Phillips|Trott|2022|p=3}}<ref>{{cite web |title=UnityDimensions—Wolfram Language Documentation |url=https://reference.wolfram.com/language/ref/UnityDimensions.html |access-date=1 July 2022 |website=reference.wolfram.com}}</ref>