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Radian
An arc of acirclewith the same length as theradiusof that circle subtends anangleof 1 radian. The circumference subtends an angle of 2πradians.
General information
Unit systemSI
Unit ofangle
Symbolrad, R[1]
Conversions
1 radin...... is equal to...
milliradians1000 mrad
turns1/2πturn
degrees180/π° ≈ 57.296°
gradians200/πgrad ≈ 63.662g

Theradian,denoted by the symbolrad,is the unit ofanglein theInternational System of Units(SI) and is the standard unit of angular measure used in many areas ofmathematics.It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius.[2]The unit was formerly anSI supplementary unitand is currently adimensionlessSI derived unit,[2]defined in the SI as 1 rad = 1[3]and expressed in terms of theSI base unitmetre(m) asrad = m/m.[4]Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.[5]

Definition

One radian is defined as the anglesubtendedfrom the center of a circle which intercepts an arc equal in length to the radius of the circle.[6]More generally, themagnitudein radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is,,whereθis the magnitude of the subtended angle in radians (= angle/rad),sis arc length, andris radius. Aright angleis exactlyradians.[7]

In the SI (and most areas of mathematics and science), what is called "angle" is actually the numerical value of the angle when the angle is expressed in radians: angle/rad. The SI "radian" is thus the numerical value of one radian when expressed in radians, hence the SI "rad" = (1 rad)/rad = 1. This has caused some confusion. The numerical value of the rotation angle when expressed in radians (i.e. 360°/rad) corresponding to one complete revolution is the length of the circumference divided by the radius, which is,or2π.Thus,2πradians is equal to 360 degrees.

The relation2πrad = 360°can be derived using the formula forarc length,.Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle,.This can be further simplified to.Multiplying both sides by360°gives360° = 2πrad.

Unit symbol

TheInternational Bureau of Weights and Measures[7]andInternational Organization for Standardization[8]specifyradas the symbol for the radian. Alternative symbols that were in use in 1909 arec(the superscript letter c, for "circular measure" ), the letter r, or a superscriptR,[1]but these variants are infrequently used, as they may be mistaken for adegree symbol(°) or a radius (r). Hence an angle of 1.2 radians would be written today as 1.2 rad; archaic notations could include 1.2 r, 1.2rad,1.2c,or 1.2R.

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, thedegree sign°is used.

Dimensional analysis

Plane angle may be defined asθ=s/r,whereθis the (numerical value of the) subtended angle in radians,sis arc length, andris radius. One SI radian corresponds to the (numerical value of the) angle expressed in radians for whichs=r,hence1 SI radian = 1 m/m= 1.[9]However,radis only to be used to express angles, not to express ratios of lengths in general.[7]A similar calculation usingthe area of a circular sectorθ= 2A/r2gives 1 SI radian as 1 m2/m2= 1.[10]The key fact is that the SI radian is adimensionless unitequal to1.In SI 2019, the SI radian is defined accordingly as1 rad = 1.[11]It is a long-established practice in mathematics and across all areas of science to make use ofrad = 1.[4][12]

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".[13]For example, an object hanging by a string from a pulley will rise or drop byy=centimeters, whereris the numerical value of the radius of the pulley when expressed in centimeters andθis the numerical value of the angle through which the pulley turns when expressed in radians. When multiplyingrbyθthe unit radian does not appear in the result. Similarly in the formula for theangular velocityof a rolling wheel,ω=v/r,radians appear in the units ofωbut not on the right hand side.[14]Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".[15]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".[16]

In 1993 theAmerican Association of Physics TeachersMetric 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 ofangle measure(rad),angular speed(rad/s),angular acceleration(rad/s2), andtorsional stiffness(N⋅m/rad), and not in the quantities oftorque(N⋅m) andangular momentum(kg⋅m2/s).[17]

At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as abase unit of measurementfor abase quantity(and dimension) of "plane angle".[18][19][20]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 thearea of a circle,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".[21]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.[20]

In particular, Quincey identifies Torrens' proposal to introduce a constantηequal to 1 inverse radian (1 rad−1) in a fashion similar to theintroduction of the constantε0.[21][a]With this change the formula for the angle subtended at the center of a circle,s=,is modified to becomes=ηrθ,and theTaylor seriesfor thesineof an angleθbecomes:[20][22] whereis the numerical value of the angle when expressed in radians. The capitalized functionSinis the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed,[22]whilesinis the traditional function onpure numberswhich assumes its argument is a dimensionless number in radians.[23]The capitalised symbolcan be denotedif it is clear that the complete form is meant.[20][24]

Current SI can be considered relative to this framework as anatural unitsystem where the equationη= 1is assumed to hold, or similarly,1 rad = 1.Thisradian conventionallows the omission ofηin mathematical formulas.[25]

Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.[26]For example, theBoostunits library defines angle units with aplane_angledimension,[27]andMathematica's unit system similarly considers angles to have an angle dimension.[28][29]

Conversions

Conversion of common angles
Turns Radians Degrees Gradians
0 turn 0 rad 0g
1/72turn π/36or𝜏/72rad ⁠5+5/9g
1/24turn π/12or𝜏/24rad 15° ⁠16+2/3g
1/16turn π/8or𝜏/16rad 22.5° 25g
1/12turn π/6or𝜏/12rad 30° ⁠33+1/3g
1/10turn π/5or𝜏/10rad 36° 40g
1/8turn π/4or𝜏/8rad 45° 50g
1/2πor 𝜏turn 1 rad approx.57.3° approx.63.7g
1/6turn π/3or𝜏/6rad 60° ⁠66+2/3g
1/5turn 2πor 𝜏/5rad 72° 80g
1/4turn π/2or𝜏/4rad 90° 100g
1/3turn 2πor 𝜏/3rad 120° ⁠133+1/3g
2/5turn 4πor 2𝜏 or α/5rad 144° 160g
1/2turn πor𝜏/2rad 180° 200g
3/4turn 3πor ρ/2or3𝜏/4rad 270° 300g
1 turn 𝜏 or 2πrad 360° 400g

Between degrees

As stated, one radian is equal to.Thus, to convert from radians to degrees, multiply magnitude in radians by.

For example:

Conversely, to convert from degrees to radians, multiply magnitude in degrees by.

For example:

Radians can be converted toturns(one turn is the angle corresponding to a revolution) by dividing the number of radians by 2π.

Between gradians

One revolution isradians, which equals oneturn,which is by definition 400gradians(400gonsor 400g). To convert from radians to gradians multiply the magnitude of the angle in radians by,and to convert from gradians to radians multiply the magnitude of the angle in gradians by.For example,

Usage

Mathematics

Some common angles, measured in radians. All the large polygons in this diagram areregular polygons.

Incalculusand most other branches of mathematics beyond practicalgeometry,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 inanalysisinvolvingtrigonometric functionscan be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simplelimitformula

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

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 thedifferential equation,the evaluation of the integraland 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, whenxis the numerical value of an angle when the angle is expressed in radians, i.ex= angle /rad, theTaylor seriesfor sinxbecomes:

Ifxwere the numerical value of the angle when expressed in degrees, i.e. the number of degrees, then the series would contain messy factors involving powers ofπ/180: ifxis the number of degrees, angle/degree, the number of radians, angle/radian, isy=πx/ 180,so

In a similar spirit, if angles are involved, mathematically important relationships between the sine and cosine functions and theexponential function(see, for example,Euler's formula) can be elegantly stated when the functions' arguments are magnitudes of 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.

Physics

The radian is widely used inphysicswhen angular measurements are required. For example,angular velocityis typically expressed in the unitradian per second(rad/s). One revolution per second corresponds to 2πradians per second.

Similarly, the unit used forangular accelerationis often radian per second per second (rad/s2).

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

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⋅2π) radians, wherenis an integer, they are considered to be inphase,whilst if the phase angle difference of two waves is (n⋅2π+π) radians, withnan 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 (forangular wavelength) or newton-metre per radian (for torsional stiffness).

Prefixes and variants

Metric prefixesfor submultiples are used with radians. Amilliradian(mrad) is a thousandth of a radian (0.001 rad), i.e.1 rad = 103mrad.There are 2π× 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under1/6283of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use bytelescopic sightmanufacturers using(stadiametric) rangefindinginreticles.Thedivergenceoflaserbeams is also usually measured in milliradians.

Theangular milis an approximation of the milliradian used byNATOand other military organizations ingunneryandtargeting.Each angular mil represents1/6400of a circle and is15/8% 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 to1/2000π;for example Sweden used the1/6300streckand the USSR used1/6000.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,10−6rad) and nanoradians (nrad,10−9rad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is thearc second,which isπ/648,000rad (around 4.8481 microradians).


SI multiples of radian (rad)
Submultiples Multiples
Value SI symbol Name Value SI symbol Name
10−1rad drad deciradian 101rad darad decaradian
10−2rad crad centiradian 102rad hrad hectoradian
10−3rad mrad milliradian 103rad krad kiloradian
10−6rad μrad microradian 106rad Mrad megaradian
10−9rad nrad nanoradian 109rad Grad gigaradian
10−12rad prad picoradian 1012rad Trad teraradian
10−15rad frad femtoradian 1015rad Prad petaradian
10−18rad arad attoradian 1018rad Erad exaradian
10−21rad zrad zeptoradian 1021rad Zrad zettaradian
10−24rad yrad yoctoradian 1024rad Yrad yottaradian
10−27rad rrad rontoradian 1027rad Rrad ronnaradian
10−30rad qrad quectoradian 1030rad Qrad quettaradian

History

Pre-20th century

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-calleddiameter partsas units, where one diameter part was1/60radian. They also used sexagesimal subunits of the diameter part.[30]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.[31]

The concept oftheradian measure is normally credited toRoger Cotes,who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book,Harmonia mensurarum.[32]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 thesemicircumference,this is as 1 to 3.141592653589 "–, and recognized its naturalness as a unit of angular measure.[33][34]

In 1765,Leonhard Eulerimplicitly adopted the radian as a unit of angle.[31]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."[35]Euler was probably the first to adopt this convention, referred to as the radian convention, which gives the simple formula for angular velocityω=v/r.As discussed in§ Dimensional analysis,the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for exampleω=v/(ηr).[25]

Prior to the termradianbecoming widespread, the unit was commonly calledcircular measureof an angle.[36]The termradianfirst appeared in print on 5 June 1873, in examination questions set byJames Thomson(brother ofLord Kelvin) atQueen's College,Belfast.He had used the term as early as 1871, while in 1869,Thomas Muir,then of theUniversity of St Andrews,vacillated between the termsrad,radial,andradian.In 1874, after a consultation with James Thomson, Muir adoptedradian.[37][38][39]The nameradianwas not universally adopted for some time after this.Longmans' School Trigonometrystill called the radiancircular measurewhen published in 1890.[40]

In 1893Alexander Macfarlanewrote "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."[41]For some reason the paper was withdrawn from the published proceedings of mathematical congress held in connection withWorld's Columbian Expositionin Chicago (acknowledged at page 167), and privately published in hisPapers on Space Analysis(1894). Macfarlane reached this idea or ratios of areas while considering the basis forhyperbolic anglewhich is analogously defined.[42]

As an SI unit

As Paul Quincey et al. writes, "the status of angles within theInternational System of Units(SI) has long been a source of controversy and confusion. "[43]In 1960, theCGPMestablished the SI and the radian was classified as a "supplementary unit" along with thesteradian.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.[44]Richard Nelson writes "This ambiguity [in the classification of the supplemental units] prompted a spirited discussion over their proper interpretation."[45]In May 1980 theConsultative Committee for Units (CCU)considered a proposal for making radians an SI base unit, using a constantα0= 1 rad,[46][25]but turned it down to avoid an upheaval to current practice.[25]

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,[45]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".[47]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".[48]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".[49]

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 theCCU Working Group on Angles and Dimensionless Quantities in the SIwas established.[50]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.[51][52]

See also

Notes

  1. ^Other proposals include the abbreviation "rad" (Brinsmade 1936), the notation(Romain 1962), and the constantsם(Brownstein 1997), ◁ (Lévy-Leblond 1998),k(Foster 2010),θC(Quincey 2021), and(Mohr et al. 2022).

References

  1. ^abHall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter VII. The General Angle [55] Signs and Limitations in Value. Exercise XV.". Written at Ann Arbor, Michigan, USA.Trigonometry.Vol. Part I: Plane Trigonometry. New York, USA:Henry Holt and Company/ Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 73.Retrieved2017-08-12.
  2. ^abInternational Bureau of Weights and Measures 2019,p. 151: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units."
  3. ^International Bureau of Weights and Measures 2019,p. 151: "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
  4. ^abInternational Bureau of Weights and Measures 2019,p. 137.
  5. ^Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Revision 3.National Aeronautics and Space Administration, Goddard Space Flight Center. 2002. p. 12.
  6. ^Protter, Murray H.; Morrey, Charles B. Jr. (1970),College Calculus with Analytic Geometry(2nd ed.), Reading:Addison-Wesley,p. APP-4,LCCN76087042
  7. ^abcInternational Bureau of Weights and Measures 2019,p. 151.
  8. ^"ISO 80000-3:2006 Quantities and Units - Space and Time".17 January 2017.
  9. ^International Bureau of Weights and Measures 2019,p. 151: "One radian corresponds to the angle for whichs=r"
  10. ^Quincey 2016,p. 844: "Also, as alluded to inMohr & Phillips 2015,the radian can be defined in terms of the areaAof a sector (A=1/2θr2), in which case it has the units m2⋅m−2."
  11. ^International Bureau of Weights and Measures 2019,p. 151: "One radian corresponds to the angle for whichs=r,thus1 rad = 1."
  12. ^Bridgman, Percy Williams (1922).Dimensional analysis.New Haven: Yale University Press.Angular amplitude of swing [...] No dimensions.
  13. ^Prando, Giacomo (August 2020)."A spectral unit".Nature Physics.16(8): 888.Bibcode:2020NatPh..16..888P.doi:10.1038/s41567-020-0997-3.S2CID225445454.
  14. ^Leonard, William J. (1999).Minds-on Physics: Advanced topics in mechanics.Kendall Hunt. p. 262.ISBN978-0-7872-5412-4.
  15. ^French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)".The Physics Teacher.30(5): 260–261.doi:10.1119/1.2343535.
  16. ^Oberhofer, E. S. (March 1992). "What happens to the 'radians'?".The Physics Teacher.30(3): 170–171.Bibcode:1992PhTea..30..170O.doi:10.1119/1.2343500.
  17. ^Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".The Physics Teacher.31(2): 84–87.Bibcode:1993PhTea..31...84A.doi:10.1119/1.2343667.
  18. ^Brinsmade 1936;Romain 1962;Eder 1982;Torrens 1986;Brownstein 1997;Lévy-Leblond 1998;Foster 2010;Mills 2016;Quincey 2021;Leonard 2021;Mohr et al. 2022
  19. ^Mohr & Phillips 2015.
  20. ^abcdQuincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI".Metrologia.53(3): 998–1002.arXiv:1604.02373.Bibcode:2016Metro..53..998Q.doi:10.1088/0026-1394/53/3/998.S2CID119294905.
  21. ^abQuincey 2016.
  22. ^abTorrens 1986.
  23. ^Mohr et al. 2022,p. 6.
  24. ^Mohr et al. 2022,pp. 8–9.
  25. ^abcdQuincey 2021.
  26. ^Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems".Metrologia.54(4): 454–460.arXiv:1705.03765.Bibcode:2017Metro..54..454Q.doi:10.1088/1681-7575/aa7160.S2CID119418270.
  27. ^Schabel, Matthias C.; Watanabe, Steven."Boost.Units FAQ – 1.79.0".boost.org.Retrieved5 May2022.Angles are treated as units
  28. ^Mohr et al. 2022,p. 3.
  29. ^"UnityDimensions—Wolfram Language Documentation".reference.wolfram.Retrieved1 July2022.
  30. ^Luckey, Paul (1953) [Translation of 1424 book]. Siggel, A. (ed.).Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi[Treatise on the Circumference of al-Kashi]. Berlin: Akademie Verlag. p. 40.
  31. ^abRoche, John J. (21 December 1998).The Mathematics of Measurement: A Critical History.Springer Science & Business Media. p. 134.ISBN978-0-387-91581-4.
  32. ^O'Connor, J. J.; Robertson, E. F. (February 2005)."Biography of Roger Cotes".The MacTutor History of Mathematics.Archived fromthe originalon 2012-10-19.Retrieved2006-04-21.
  33. ^Cotes, Roger (1722)."Editoris notæ ad Harmoniam mensurarum".In Smith, Robert (ed.).Harmonia mensurarum(in Latin). Cambridge, England. pp. 94–95.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.[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.]
  34. ^Gowing, Ronald (27 June 2002).Roger Cotes - Natural Philosopher.Cambridge University Press.ISBN978-0-521-52649-4.
  35. ^Euler, Leonhard.Theoria Motus Corporum Solidorum seu Rigidorum[Theory of the motion of solid or rigid bodies](PDF)(in Latin). Translated by Bruce, Ian. Definition 6, paragraph 316.
  36. ^Isaac Todhunter,Plane Trigonometry: For the Use of Colleges and Schools,p. 10,Cambridge and London: MacMillan, 1864OCLC500022958
  37. ^Cajori, Florian(1929).History of Mathematical Notations.Vol. 2. Dover Publications. pp.147–148.ISBN0-486-67766-4.
  38. ^
  39. ^Miller, Jeff (Nov 23, 2009)."Earliest Known Uses of Some of the Words of Mathematics".RetrievedSep 30,2011.
  40. ^Frederick Sparks,Longmans' School Trigonometry,p. 6, London: Longmans, Green, and Co., 1890OCLC877238863(1891 edition)
  41. ^A. Macfarlane (1893) "On the definitions of the trigonometric functions", page 9,link at Internet Archive
  42. ^Geometry/Unified Anglesat Wikibooks
  43. ^Quincey, Paul; Mohr, Peter J; Phillips, William D (1 August 2019). "Angles are inherently neither length ratios nor dimensionless".Metrologia.56(4): 043001.arXiv:1909.08389.Bibcode:2019Metro..56d3001Q.doi:10.1088/1681-7575/ab27d7.S2CID198428043.
  44. ^Le Système international d'unités(PDF)(in French), 1970, p. 12,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.[For some units of the SI, the CGPM still hasn't yet decided whether they are base units or derived units.]
  45. ^abNelson, Robert A. (March 1984). "The supplementary units".The Physics Teacher.22(3): 188–193.Bibcode:1984PhTea..22..188N.doi:10.1119/1.2341516.
  46. ^Report of the 7th meeting(PDF)(in French), Consultative Committee for Units, May 1980, pp. 6–7
  47. ^International Bureau of Weights and Measures 2019,pp. 174–175.
  48. ^International Bureau of Weights and Measures 2019,p. 179.
  49. ^Kalinin, Mikhail I (1 December 2019). "On the status of plane and solid angles in the International System of Units (SI)".Metrologia.56(6): 065009.arXiv:1810.12057.Bibcode:2019Metro..56f5009K.doi:10.1088/1681-7575/ab3fbf.S2CID53627142.
  50. ^Consultative Committee for Units(11–12 June 2013).Report of the 21st meeting to the International Committee for Weights and Measures(Report). pp. 18–20.
  51. ^Consultative Committee for Units(21–23 September 2021).Report of the 25th meeting to the International Committee for Weights and Measures(Report). pp. 16–17.
  52. ^"CCU Task Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB)".BIPM.Retrieved26 June2022.
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