Radian#Unit symbol

One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals the radius of the circle.{{ 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 }} More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, $\backslash theta\; =\; \backslash frac\{s\}\{r\}$, 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 $\backslash frac\{\backslash pi\}\{2\}$ 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 $\backslash frac\; \{2\backslash pi\; r\}\{r\}$, or {{math|2π}}. Thus, {{math|2π}} radians is equal to 360 degrees. The relation {{math|1=2π rad = 360°}} can be derived using the formula for arc length, $\backslash ell\_\{\backslash text\{arc\}\}=2\backslash pi\; r\backslash left(\backslash tfrac\{\backslash theta\}\{360^\{\backslash circ\}\}\backslash right)$. Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle, $1=2\backslash pi\backslash left(\backslash tfrac\{1\backslash text\{\; rad\}\}\{360^\{\backslash circ\}\}\backslash right)$. This can be further simplified to $1=\backslash tfrac\{2\backslash pi\backslash text\{\; rad\}\}\{360^\{\backslash circ\}\}$. Multiplying both sides by {{math|360°}} gives {{math|360° {{=}} 2π rad}}.

The International Bureau of Weights and Measures{{sfn|International Bureau of Weights and Measures|2019|p=151}} and International Organization for Standardization{{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}} specify rad as the symbol for the radian. Alternative symbols that were in use in 1909 are ^c (the superscript letter c, for "circular measure"), the letter r, or a superscript {{sup|R}}, 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 rad; archaic notations include 1.2 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.

{{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 m/m}} = 1.{{harvnb|International Bureau of Weights and Measures|2019|p=151}}: "One radian corresponds to the angle for which {{math|1=s = r}}" 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 the area of a circular sector {{math|1=θ = 2A/r²}} gives 1 radian as 1 m²/m² = 1.{{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 r²}}), in which case it has the units m²⋅m⁻²." 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}}.{{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}}." 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}}{{cite book |last1=Bridgman |first1=Percy Williams |url=https://archive.org/details/dimensionalanaly00bridrich/page/n13/mode/2up |title=Dimensional analysis |date=1922 |publisher=New Haven : Yale University Press |quote=Angular amplitude of swing [...] No dimensions.}}

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".{{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 }} 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.{{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}} Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".{{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}} 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".{{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}}

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/s²), and torsional stiffness (N⋅m/rad), and not in the quantities of torque (N⋅m) and angular momentum (kg⋅m²/s).{{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}}

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".{{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}}{{sfn|Mohr|Phillips|2015}}{{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}} 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=πr²}}. 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.

In particular, Quincey identifies Torrens' proposal to introduce a constant {{mvar|η}} equal to 1 inverse radian (1 rad⁻¹) in a fashion similar to the introduction of the constant ε₀.{{sfn|Quincey|2016}}{{efn|Other proposals include the abbreviation "rad" {{harv|Brinsmade|1936}}, the notation $\backslash langle\; \backslash theta\; \backslash rangle$ {{harv|Romain|1962}}, and the constants ם {{harv|Brownstein|1997}}, ◁ {{harv|Lévy-Leblond|1998}}, k {{harv|Foster|2010}}, θ_C {{harv|Quincey|2021}}, and $\{\backslash cal\; C\}\; =\; \backslash frac\{2\backslash pi\}\{\backslash Theta\}$ {{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:{{sfn|Torrens|1986}}

$$\backslash operatorname\{Sin\}\; \backslash theta\; =\; \backslash sin\; \backslash \; x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash frac\{x^7\}\{7!\}\; +\; \backslash cdots\; =\; \backslash eta\; \backslash theta\; -\; \backslash frac\{(\backslash eta\; \backslash theta)^3\}\{3!\}\; +\; \backslash frac\{(\backslash eta\; \backslash theta)^5\}\{5!\}\; -\; \backslash frac\{(\backslash eta\; \backslash theta)^7\}\{7!\}\; +\; \backslash cdots\; ,$$

where $x\; =\; \backslash eta\; \backslash theta\; =\; \backslash theta/\backslash text\{rad\}$ 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 numbers which assumes its argument is a dimensionless number in radians.{{sfn|Mohr|Shirley|Phillips|Trott|2022|p=6}} The capitalised symbol $\backslash operatorname\{Sin\}$ can be denoted $\backslash sin$ if it is clear that the complete form is meant.{{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.{{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}} For example, the Boost units library defines angle units with a plane_angle dimension,{{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}} and Mathematica's unit system similarly considers angles to have an angle dimension.{{sfn|Mohr|Shirley|Phillips|Trott|2022|p=3}}{{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}}

{{Table of angles|float=right}}

As stated, one radian is equal to $\{180^\backslash circ\}/\{\backslash pi\}$. Thus, to convert from radians to degrees, multiply by $\{180^\backslash circ\}/\{\backslash pi\}$.

:$\backslash text\{angle\; in\; degrees\}\; =\; \backslash text\{angle\; in\; radians\}\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}$

For example:

:$1\; \backslash text\{\; rad\}\; =\; 1\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; 57.2958^\backslash circ$

:$2.5\; \backslash text\{\; rad\}\; =\; 2.5\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; 143.2394^\backslash circ$

:$\backslash frac\; \{\backslash pi\}\; \{3\}\; \backslash text\{\; rad\}\; =\; \backslash frac\; \{\backslash pi\}\; \{3\}\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; =\; 60^\backslash circ$

Conversely, to convert from degrees to radians, multiply by $\{\backslash pi\}/\{180\}\; \backslash text\{\; rad\}$.

:$\backslash text\{angle\; in\; radians\}\; =\; \backslash text\{angle\; in\; degrees\}\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180\}\; \backslash text\{\; rad\}$

For example:

:$1^\backslash circ\; =\; 1\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180\}\; \backslash text\{\; rad\}\backslash approx\; 0.0175\; \backslash text\{\; rad\}$

$23^\backslash circ\; =\; 23\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180\}\; \backslash text\{\; rad\}\backslash approx\; 0.4014\; \backslash text\{\; rad\}$

Radians can be converted to turns (one turn is the angle corresponding to a revolution) by dividing the number of radians by 2{{pi}}.

One revolution is $2\backslash pi$ radians, which equals one turn, which is by definition 400 gradians (400 gons or 400^g). To convert from radians to gradians multiply by $200^\backslash text\{g\}/\backslash pi$, and to convert from gradians to radians multiply by $\backslash pi/200\; \backslash text\{\; rad\}$. For example,

:$1.2\; \backslash text\{\; rad\}\; =\; 1.2\; \backslash cdot\; \backslash frac\; \{200^\backslash text\{g\}\}\; \{\backslash pi\}\; \backslash approx\; 76.3944^\backslash text\{g\}$

:$50^\backslash text\{g\}\; =\; 50\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{200\}\; \backslash text\{\; rad\}\backslash approx\; 0.7854\; \backslash text\{\; rad\}$

File:Radian-common.svgs.]]

In calculus and most other branches of mathematics beyond practical geometry, angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results.

Results in analysis involving trigonometric functions can be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

:$\backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{\backslash sin\; h\}\{h\}=1,$

which is the basis of many other identities in mathematics, including

:$\backslash frac\{d\}\{dx\}\; \backslash sin\; x\; =\; \backslash cos\; x$

:$\backslash frac\{d^2\}\{dx^2\}\; \backslash sin\; x\; =\; -\backslash sin\; x.$

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation $\backslash tfrac\{d^2\; y\}\{dx^2\}\; =\; -y$, the evaluation of the integral $\backslash textstyle\backslash int\; \backslash frac\{dx\}\{1+x^2\},$ and so on). In all such cases, it is appropriate that the arguments of the functions are treated as (dimensionless) numbers—without any reference to angles.

The trigonometric functions of angles also have simple and elegant series expansions when radians are used. For example, when x is the angle expressed in radians, the Taylor series for sin x becomes:

:$\backslash sin\; x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash frac\{x^7\}\{7!\}\; +\; \backslash cdots\; .$

If y were the angle x but expressed in degrees, i.e. {{nowrap|1=y = {{pi}}x / 180}}, then the series would contain messy factors involving powers of {{pi}}/180:

:$\backslash sin\; y\; =\; \backslash frac\{\backslash pi\}\{180\}\; x\; -\; \backslash left\; (\backslash frac\{\backslash pi\}\{180\}\; \backslash right\; )^3\backslash \; \backslash frac\{x^3\}\{3!\}\; +\; \backslash left\; (\backslash frac\{\backslash pi\}\{180\}\; \backslash right\; )^5\backslash \; \backslash frac\{x^5\}\{5!\}\; -\; \backslash left\; (\backslash frac\{\backslash pi\}\{180\}\; \backslash right\; )^7\backslash \; \backslash frac\{x^7\}\{7!\}\; +\; \backslash cdots\; .$

In a similar spirit, if angles are involved, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) can be elegantly stated when the functions' arguments are angles expressed in radians (and messy otherwise). More generally, in complex-number theory, the arguments of these functions are (dimensionless, possibly complex) numbers—without any reference to physical angles at all.

The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically expressed in the unit radian per second (rad/s). One revolution per second corresponds to 2{{pi}} radians per second.

Similarly, the unit used for angular acceleration is often radian per second per second (rad/s²).

For the purpose of dimensional analysis, the units of angular velocity and angular acceleration are s⁻¹ and s⁻² respectively.

Likewise, the phase angle difference of two waves can also be expressed using the radian as the unit. For example, if the phase angle difference of two waves is (n⋅2{{pi}}) radians, where n is an integer, they are considered to be in phase, whilst if the phase angle difference of two waves is ({{nowrap|n⋅2{{pi}} + {{pi}}}}) radians, with n an integer, they are considered to be in antiphase.

A unit of reciprocal radian or inverse radian (rad⁻¹) is involved in derived units such as meter per radian (for angular wavelength) or newton-metre per radian (for torsional stiffness).

Metric prefixes for submultiples are used with radians. A milliradian (mrad) is a thousandth of a radian (0.001 rad), i.e. {{nowrap|1=1 rad = 10³ mrad}}. There are 2{{pi}} × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under {{sfrac|1|6283}} of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

The angular mil is an approximation of the milliradian used by NATO and other military organizations in gunnery and targeting. Each angular mil represents {{sfrac|1|6400}} of a circle and is {{sfrac|15|8}}% or 1.875% smaller than the milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to {{sfrac|1|2000{{pi}}}}; for example Sweden used the {{sfrac|1|6300}} streck and the USSR used {{sfrac|1|6000}}. Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad, {{val|e=-6|u=rad}}) and nanoradians (nrad, {{val|e=-9|u=rad}}) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is the arc second, which is {{sfrac|{{pi}}|648,000}} rad (around 4.8481 microradians).

{{SI multiples

| unit=radian

| symbol=rad

| xm=milliradian

}}

The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example, al-Kashi (c. 1400) used so-called diameter parts as units, where one diameter part was {{sfrac|1|60}} radian. They also used sexagesimal subunits of the diameter part.{{cite book|first=Paul|last= Luckey|editor-first=A. |editor-last=Siggel|location=Berlin|publisher= Akademie Verlag| orig-year=Translation of 1424 book|year=1953| title=Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi|trans-title=Treatise on the Circumference of al-Kashi| number=6|pages= 40}} Newton in 1672 spoke of "the angular quantity of a body's circular motion", but used it only as a relative measure to develop an astronomical algorithm.{{cite book |last1=Roche |first1=John J. |title=The Mathematics of Measurement: A Critical History |date=21 December 1998 |publisher=Springer Science & Business Media |isbn=978-0-387-91581-4 |page=134 |url=https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA134 |language=en}}

The concept of the radian measure is normally credited to Roger Cotes, who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, Harmonia mensurarum.{{cite web |url = http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html |title = Biography of Roger Cotes |work = The MacTutor History of Mathematics |date = February 2005 |last1 = O'Connor |first1 = J. J. |first2 = E. F. |last2 = Robertson |access-date = 2006-04-21 |archive-date = 2012-10-19 |archive-url = https://web.archive.org/web/20121019161705/http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html |url-status = dead }} In a chapter of editorial comments, Smith gave what is probably the first published calculation of one radian in degrees, citing a note of Cotes that has not survived. Smith described the radian in everything but name – "Now this number is equal to 180 degrees as the radius of a circle to the semicircumference, this is as 1 to 3.141592653589" –, and recognized its naturalness as a unit of angular measure.{{cite book |last1=Cotes |first1=Roger |title=Harmonia mensurarum |date=1722 |editor-first=Robert|editor-last=Smith|location=Cambridge, England|chapter=Editoris notæ ad Harmoniam mensurarum |pages=94–95 |chapter-url=https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA2-PA95 |language=la|quote=In Canone Logarithmico exhibetur Systema quoddam menfurarum numeralium, quæ Logarithmi dicuntur: atque hujus systematis Modulus is est Logarithmus, qui metitur Rationem Modularem in Corol. 6. definitam. Similiter in Canone Trigonometrico finuum & tangentium, exhibetur Systema quoddam menfurarum numeralium, quæ Gradus appellantur: atque hujus systematis Modulus is est Numerus Graduum, qui metitur Angulum Modularem modo definitun, hoc est, qui continetur in arcu Radio æquali. Eft autem hic Numerus ad Gradus 180 ut Circuli Radius ad Semicircuinferentiam, hoc eft ut 1 ad 3.141592653589 &c. Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. Cujus Reciprocus eft 0.0174532925 &c. Hujus moduli subsidio (quem in chartula quadam Auctoris manu descriptum inveni) commodissime computabis mensuras angulares, queinadmodum oftendam in Nota III.|trans-quote=In the Logarithmic Canon there is presented a certain system of numerical measures called Logarithms: and the Modulus of this system is the Logarithm, which measures the Modular Ratio as defined in Corollary 6. Similarly, in the Trigonometrical Canon of sines and tangents, there is presented a certain system of numerical measures called Degrees: and the Modulus of this system is the Number of Degrees which measures the Modular Angle defined in the manner defined, that is, which is contained in an equal Radius arc. Now this Number is equal to 180 Degrees as the Radius of a Circle to the Semicircumference, this is as 1 to 3.141592653589 &c. Hence the Modulus of the Trigonometric Canon will be 57.2957795130 &c. Whose Reciprocal is 0.0174532925 &c. With the help of this modulus (which I found described in a note in the hand of the Author) you will most conveniently calculate the angular measures, as mentioned in Note III.}}{{cite book |last1=Gowing |first1=Ronald |title=Roger Cotes - Natural Philosopher |date=27 June 2002 |publisher=Cambridge University Press |isbn=978-0-521-52649-4 |url=https://books.google.com/books?id=I2Cy4wjj1soC&pg=PA39}}

In 1765, Leonhard Euler implicitly adopted the radian as a unit of angle. Specifically, Euler defined angular velocity as "The angular speed in rotational motion is the speed of that point, the distance of which from the axis of gyration is expressed by one."{{cite book |last1=Euler |first1=Leonhard |translator1-last=Bruce |translator1-first=Ian |title=Theoria Motus Corporum Solidorum seu Rigidorum|trans-title= Theory of the motion of solid or rigid bodies|language=latin|at=Definition 6, paragraph 316|url=http://www.17centurymaths.com/contents/euler/mechvol3/tmvol1ch2tr.pdf#page=3}} Euler was probably the first to adopt this convention, referred to as the radian convention, which gives the simple formula for angular velocity {{math|1=ω = v/r}}. As discussed in {{section link|#Dimensional analysis}}, the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for example {{math|1=ω = v/(ηr)}}.{{sfn|Quincey|2021}}

Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.Isaac Todhunter, Plane Trigonometry: For the Use of Colleges and Schools, [https://books.google.com/books?id=bo5FAAAAcAAJ&pg=PA10 p. 10], Cambridge and London: MacMillan, 1864 {{OCLC|500022958}} The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between the terms rad, radial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian.{{cite book| author-link=Florian Cajori| first=Florian| last=Cajori| year=1929| title=History of Mathematical Notations| volume=2| pages=[https://archive.org/details/historyofmathema00cajo_0/page/147 147–148]| publisher=Dover Publications| isbn=0-486-67766-4| url-access=registration| url=https://archive.org/details/historyofmathema00cajo_0/page/147}}

• {{cite journal| journal=Nature| year=1910| volume= 83| pages=156|doi=10.1038/083156a0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2110|bibcode = 1910Natur..83..156M | s2cid=3958702| url=https://zenodo.org/record/1429528| doi-access=free}}
• {{cite journal| journal=Nature| year=1910| volume= 83| pages=217|doi=10.1038/083217c0| title=The Term "Radian" in Trigonometry| last1=Thomson| first1=James| issue=2112|bibcode = 1910Natur..83..217T | s2cid=3980250| url=https://zenodo.org/record/1429530| doi-access=free}}
• {{cite journal| journal=Nature| year=1910| volume= 83| pages=459–460|doi=10.1038/083459d0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2120|bibcode = 1910Natur..83..459M | s2cid=3971449| url=https://zenodo.org/record/1429528}}{{cite web|url=http://jeff560.tripod.com/r.html|date=Nov 23, 2009| access-date=Sep 30, 2011|last=Miller|first=Jeff |title= Earliest Known Uses of Some of the Words of Mathematics}} The name radian was not universally adopted for some time after this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.Frederick Sparks, Longmans' School Trigonometry, p. 6, London: Longmans, Green, and Co., 1890 {{oclc|877238863}} (1891 edition)

In 1893 Alexander Macfarlane wrote "the true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of a sector to the square on the radius."A. Macfarlane (1893) "On the definitions of the trigonometric functions", page 9, [https://archive.org/details/principlesalgeb01macfgoog/page/n138/mode/2up link at Internet Archive] However, the paper was withdrawn from the published proceedings of mathematical congress held in connection with World's Columbian Exposition in Chicago (acknowledged at page 167), and privately published in his Papers on Space Analysis (1894). Macfarlane reached this idea or ratios of areas while considering the basis for hyperbolic angle which is analogously defined.{{wikibooks inline|Geometry/Unified Angles}}

{{see also|#Dimensional analysis}}

As Paul Quincey et al. write, "the status of angles within the International System of Units (SI) has long been a source of controversy and confusion."{{cite journal |last1=Quincey |first1=Paul |last2=Mohr |first2=Peter J. |last3=Phillips |first3=William D. |title=Angles are inherently neither length ratios nor dimensionless |journal=Metrologia |date=1 August 2019 |volume=56 |issue=4 |pages=043001 |doi=10.1088/1681-7575/ab27d7|arxiv=1909.08389|bibcode=2019Metro..56d3001Q |s2cid=198428043}} In 1960, the General Conference on Weights and Measures (CGPM) established the SI and the radian was classified as a "supplementary unit" along with the steradian. This special class was officially regarded "either as base units or as derived units", as the CGPM could not reach a decision on whether the radian was a base unit or a derived unit.{{citation|url=https://www.bipm.org/documents/20126/41483022/si_brochure_1.pdf/26bf12f9-1da7-e6c2-254b-c50fb978c01b|page=12|title=Le Système international d'unités|date=1970|language=fr|quote=Pour quelques unités du Système International, la Conférence Générale n'a pas ou n'a pas encore décidé s'il s'agit d'unités de base ou bien d'unités dérivées.|trans-quote=For some units of the SI, the CGPM still hasn't yet decided whether they are base units or derived units.}} Richard Nelson writes "This ambiguity [in the classification of the supplemental units] prompted a spirited discussion over their proper interpretation." In May 1980 the Consultative Committee for Units (CCU) considered a proposal for making radians an SI base unit, using a constant {{math|1=α₀ = 1 rad}},{{citation |title=Report of the 7th meeting |publisher=Consultative Committee for Units |date=May 1980 |pages=6–7 |url=https://www.bipm.org/documents/20126/30132777/cc-publication-ID-422/cd7e3b5f-471c-4e5c-e4e1-eb998acb155f|language=fr}}{{sfn|Quincey|2021}} but turned it down to avoid an upheaval to current practice.{{sfn|Quincey|2021}}

In October 1980 the CGPM decided that supplementary units were dimensionless derived units for which the CGPM allowed the freedom of using them or not using them in expressions for SI derived units,{{cite journal |last1=Nelson |first1=Robert A. |title=The supplementary units |journal=The Physics Teacher |date=March 1984 |volume=22 |issue=3 |pages=188–193 |doi=10.1119/1.2341516|bibcode=1984PhTea..22..188N }} on the basis that "[no formalism] exists which is at the same time coherent and convenient and in which the quantities plane angle and solid angle might be considered as base quantities" and that "[the possibility of treating the radian and steradian as SI base units] compromises the internal coherence of the SI based on only seven base units".{{sfn|International Bureau of Weights and Measures|2019|pp=174-175}} In 1995 the CGPM eliminated the class of supplementary units and defined the radian and the steradian as "dimensionless derived units, the names and symbols of which may, but need not, be used in expressions for other SI derived units, as is convenient".{{sfn|International Bureau of Weights and Measures|2019|p=179}} Mikhail Kalinin writing in 2019 has criticized the 1980 CGPM decision as "unfounded" and says that the 1995 CGPM decision used inconsistent arguments and introduced "numerous discrepancies, inconsistencies, and contradictions in the wordings of the SI".{{cite journal |last1=Kalinin |first1=Mikhail I |title=On the status of plane and solid angles in the International System of Units (SI) |journal=Metrologia |date=1 December 2019 |volume=56 |issue=6 |pages=065009 |doi=10.1088/1681-7575/ab3fbf|arxiv=1810.12057|bibcode=2019Metro..56f5009K |s2cid=53627142 }}

At the 2013 meeting of the CCU, Peter Mohr gave a presentation on alleged inconsistencies arising from defining the radian as a dimensionless unit rather than a base unit. CCU President Ian M. Mills declared this to be a "formidable problem" and the CCU Working Group on Angles and Dimensionless Quantities in the SI was established.{{cite report|author=Consultative Committee for Units|author-link=Consultative Committee for Units|title=Report of the 21st meeting to the International Committee for Weights and Measures |date=11–12 June 2013|url=https://www.bipm.org/documents/20126/31581067/21st+meeting/99596189-a3f2-bc60-46bf-f7828b980a5c|pages=18–20}} The CCU met in 2021, but did not reach a consensus. A small number of members argued strongly that the radian should be a base unit, but the majority felt the status quo was acceptable or that the change would cause more problems than it would solve. A task group was established to "review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.{{cite report|author=Consultative Committee for Units|author-link=Consultative Committee for Units|title=Report of the 25th meeting to the International Committee for Weights and Measures |date=21–23 September 2021|url=https://www.bipm.org/documents/20126/68201438/CCU25.pdf/ab1be833-d656-ae2e-36fa-feb04948ed4c#page=16|pages=16–17}}{{cite web |title=CCU Task Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB) |url=https://www.bipm.org/en/committees/cc/ccu/wg/ccu-tg-adqsib |publisher=BIPM |access-date=26 June 2022}}

• Angular frequency
• Minute and second of arc
• Steradian, a higher-dimensional analog of the radian which measures solid angle
• Trigonometry

{{Notelist}}

{{Reflist|refs=

{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=January 1909 |location=Ann Arbor, Michigan, USA |chapter=Chapter VII. The General Angle [55] Signs and Limitations in Value. Exercise XV. |publisher=Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA |publication-place=New York, USA |page=73 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n88/mode/1up |access-date=2017-08-12 }}

}}

{{refbegin}}

• {{citation |title=The International System of Units (SI) |author=International Bureau of Weights and Measures |author-link=New SI |date=20 May 2019|edition=9th |isbn=978-92-822-2272-0 |url=https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf#page=37|archive-url=https://web.archive.org/web/20210508153809/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf |archive-date=8 May 2021|url-status=live}}
• {{cite journal |last1=Brinsmade |first1=J. B. |title=Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly |journal=American Journal of Physics |date=December 1936 |volume=4 |issue=4 |pages=175–179 |doi=10.1119/1.1999110|bibcode=1936AmJPh...4..175B }}
• {{cite journal |last1=Romain |first1=Jacques E. |title=Angle as a fourth fundamental quantity |journal=Journal of Research of the National Bureau of Standards Section B |date=July 1962 |volume=66B |issue=3 |pages=97 |doi=10.6028/jres.066B.012|doi-access=free}}
• {{cite journal |last1=Eder |first1=W E |title=A Viewpoint on the Quantity "Plane Angle" |journal=Metrologia |date=January 1982 |volume=18 |issue=1 |pages=1–12 |doi=10.1088/0026-1394/18/1/002|bibcode=1982Metro..18....1E |s2cid=250750831 }}
• {{cite journal |last1=Torrens |first1=A B |title=On Angles and Angular Quantities |journal=Metrologia |date=1 January 1986 |volume=22 |issue=1 |pages=1–7 |doi=10.1088/0026-1394/22/1/002|bibcode=1986Metro..22....1T |s2cid=250801509 }}
• {{cite journal |last1=Brownstein |first1=K. R. |title=Angles—Let's treat them squarely |journal=American Journal of Physics |date=July 1997 |volume=65 |issue=7 |pages=605–614 |doi=10.1119/1.18616|bibcode=1997AmJPh..65..605B |doi-access=free }}
• {{cite journal |last1=Lévy-Leblond |first1=Jean-Marc |title=Dimensional angles and universal constants |journal=American Journal of Physics |date=September 1998 |volume=66 |issue=9 |pages=814–815 |doi=10.1119/1.18964 |bibcode=1998AmJPh..66..814L |url=https://www.researchgate.net/publication/253371022}}
• {{cite journal |last1=Foster |first1=Marcus P |title=The next 50 years of the SI: a review of the opportunities for the e-Science age |journal=Metrologia |date=1 December 2010 |volume=47 |issue=6 |pages=R41–R51 |doi=10.1088/0026-1394/47/6/R01 |s2cid=117711734 |url=https://publications.csiro.au/rpr/download?pid=csiro:EP11794&dsid=DS4}}
• {{cite journal |last1=Mohr |first1=Peter J |last2=Phillips |first2=William D |title=Dimensionless units in the SI |journal=Metrologia |date=1 February 2015 |volume=52 |issue=1 |pages=40–47 |doi=10.1088/0026-1394/52/1/40|arxiv=1409.2794 |bibcode=2015Metro..52...40M |doi-access=free}}
• {{cite journal |last1=Quincey |first1=Paul |title=The range of options for handling plane angle and solid angle within a system of units |journal=Metrologia |date=1 April 2016 |volume=53 |issue=2 |pages=840–845 |doi=10.1088/0026-1394/53/2/840|bibcode=2016Metro..53..840Q |s2cid=125438811 }}
• {{cite journal |last1=Mills |first1=Ian |title=On the units radian and cycle for the quantity plane angle |journal=Metrologia |date=1 June 2016 |volume=53 |issue=3 |pages=991–997 |doi=10.1088/0026-1394/53/3/991|bibcode=2016Metro..53..991M |s2cid=126032642 }}
• {{cite journal |last1=Quincey |first1=Paul |title=Angles in the SI: a detailed proposal for solving the problem |journal=Metrologia |date=1 October 2021 |volume=58 |issue=5 |pages=053002 |doi=10.1088/1681-7575/ac023f |arxiv=2108.05704 |bibcode=2021Metro..58e3002Q |s2cid=236547235 }}
• {{cite journal |last1=Leonard |first1=B P |title=Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI) |journal=Metrologia |date=1 October 2021 |volume=58 |issue=5 |pages=052001 |doi=10.1088/1681-7575/abe0fc|bibcode=2021Metro..58e2001L |s2cid=234036217 }}
• {{cite journal |last1=Mohr |first1=Peter J |last2=Shirley |first2=Eric L |last3=Phillips |first3=William D |last4=Trott |first4=Michael |title=On the dimension of angles and their units |journal=Metrologia |date=23 June 2022 |volume=59 |issue=5 |page=053001 |doi=10.1088/1681-7575/ac7bc2|doi-access=free|arxiv=2203.12392|bibcode=2022Metro..59e3001M }}

{{refend}}

{{Wikibooks|Trigonometry/Radian and degree measures}}

{{Wiktionary|radian}}

• {{Commons category-inline}}

{{SI units}}

Category:Natural units

Category:SI derived units

Category:Pi

Category:Units of plane angle