5678:[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.]
4651:
quantities plane angle and solid angle might be considered as base quantities" and that " compromises the internal coherence of the SI based on only seven base units". 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". 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".
5676:
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.
5994:
7141:
3114:
56:
6800:
1191:
888:. 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". 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.
3846:
942:
4633:. 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. Richard Nelson writes "This ambiguity prompted a spirited discussion over their proper interpretation." In May 1980 the
395:
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
5675:
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
4659:
was established. 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
4484:
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." Euler was probably the first to adopt this convention, referred
833:
but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". 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
4594:
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." For some reason the paper was withdrawn from the published proceedings of mathematical congress held in connection with
4473:. 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
3641:
3859:) 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.
4650:
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, on the basis that " exists which is at the same time coherent and convenient and in which the
4053:% 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
2715:
2410:
784:
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". For example, an object hanging by a string from a pulley will rise or drop by
2624:
4654:
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
1186:{\displaystyle \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 ,}
3035:
3098:
2847:
2545:
673:
as the symbol for the radian. Alternative symbols that were in use in 1909 are (the superscript letter c, for "circular measure"), the letter r, or a superscript , but these variants are infrequently used, as they may be mistaken for a
3604:
2783:
2480:
3841:{\displaystyle \sin y={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}
524:
877:(and dimension) of "plane angle". 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
4457:
radian. They also used sexagesimal subunits of the diameter part. 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.
396:
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
3347:
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
587:
3209:
841:
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
639:
3466:
3342:
4809:
1236:
2672:
3411:
3268:
2664:
2365:
4732:
347:
2968:
2551:
2357:
2318:
1268:
2935:
429:
4839:
Hall, Arthur Graham; Frink, Fred
Goodrich (January 1909). "Chapter VII. The General Angle Signs and Limitations in Value. Exercise XV.". Written at Ann Arbor, Michigan, USA.
390:
681:
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
2976:
3041:
2790:
2486:
2888:
1288:
3493:
2726:
3900:
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 (
2421:
278:. 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. The unit was formerly an
6073:
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.
3133:, angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results.
3613:
were 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 of
3468:
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.
4505:, the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for example
6709:
Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the
International System of Units (SI)".
678:(°) 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, 1.2, 1.2, or 1.2.
4107:. 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).
463:
4872:, p. 151: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units."
6301:
666:
5636:
529:
6124:
662:
7116:
6242:
6063:
4660:"review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.
4626:
5521:
6010:
Quincey, Paul; Mohr, Peter J; Phillips, William D (1 August 2019). "Angles are inherently neither length ratios nor dimensionless".
3144:
can be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple
1319:
Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal. For example, the
838:
6224:
4118:) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is the
3851:
In a similar spirit, if angles are involved, mathematically important relationships between the sine and cosine functions and the
3154:
300:. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.
855:
898:
592:
3471:
The trigonometric functions of angles also have simple and elegant series expansions when radians are used. For example, when
6295:
5700:
5619:
5158:
1242:
is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while
5236:
Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".
3416:
3274:
6167:
Kalinin, Mikhail I (1 December 2019). "On the status of plane and solid angles in the
International System of Units (SI)".
4772:
4469:, who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book,
1196:
6588:
Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units".
6832:
5341:
Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI".
2710:{\displaystyle {\text{angle in radians}}={\text{magnitude of angle in degrees}}\cdot {\frac {\pi }{180}}{\text{ rad}}}
5785:
526:. Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle,
312:
from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the
7022:
6238:
6220:
6075:[For some units of the SI, the CGPM still hasn't yet decided whether they are base units or derived units.]
4634:
3354:
3220:
2405:{\displaystyle {\text{angle in degrees}}={\text{magnitude of angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}
4596:
7126:
5717:
3937:
A unit of reciprocal radian or inverse radian (rad) is involved in derived units such as meter per radian (for
2632:
803:
is the numerical value of the angle through which the pulley turns when expressed in radians. When multiplying
748:
is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using
89:
6312:
Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their
Pedagogic Value When Introduced Explicitly".
5997:
2619:{\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}
316:
in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is,
7180:
6841:
6282:
4711:
4622:
4544:
4433:
The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example,
271:
6511:
319:
5981:
5079:
2940:
2323:
2284:
4840:
3982:
of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by
1253:
730:
is radius. One SI radian corresponds to the (numerical value of the) angle expressed in radians for which
4552:
2905:
399:
368:
4849:/ Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 73
7170:
3987:
7080:
5644:
5467:
Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems".
4674:
4556:
4536:
870:
5561:
4477:, this is as 1 to 3.141592653589" –, and recognized its naturalness as a unit of angular measure.
3030:{\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}
777:. It is a long-established practice in mathematics and across all areas of science to make use of
7121:
6804:
6662:
Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem".
5666:
4110:
Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad,
4018:
3093:{\displaystyle 50^{\text{g}}=50\cdot {\frac {\pi }{200}}{\text{ rad}}\approx 0.7854{\text{ rad}}}
1320:
313:
38:
17:
5777:
5770:
7165:
6825:
6136:
4914:. National Aeronautics and Space Administration, Goddard Space Flight Center. 2002. p. 12.
4846:
3141:
3137:
2902:
or 400). To convert from radians to gradians multiply the magnitude of the angle in radians by
2842:{\displaystyle 23^{\circ }=23\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.4014{\text{ rad}}}
2540:{\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}
869:
At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a
5609:
5148:
6256:
5745:
5690:
4951:
4949:
4947:
4909:
4892:
4890:
3599:{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}
3349:
2937:, and to convert from gradians to radians multiply the magnitude of the angle in gradians by
2778:{\displaystyle 1^{\circ }=1\cdot {\frac {\pi }{180}}{\text{ rad}}\approx 0.0175{\text{ rad}}}
279:
2855:(one turn is the angle corresponding to a revolution) by dividing the number of radians by 2
6769:
6718:
6681:
6634:
6597:
6566:
6490:
6455:
6414:
6377:
6321:
6186:
6148:
6098:
6029:
5902:
5859:
5816:
5486:
5360:
5245:
5210:
5111:
4591:
3894:
3887:
3852:
2870:
1273:
851:
31:
4944:
4887:
4629:
established the SI and the radian was classified as a "supplementary unit" along with the
2475:{\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}
8:
7106:
6257:"CCU Task Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB)"
3145:
6773:
6722:
6685:
6638:
6625:
Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle".
6601:
6570:
6494:
6459:
6418:
6381:
6325:
6190:
6102:
6033:
5906:
5863:
5820:
5490:
5364:
5249:
5214:
5115:
4603:(1894). Macfarlane reached this idea or ratios of areas while considering the basis for
1250:
which assumes its argument is a dimensionless number in radians. The capitalised symbol
1238:
is the numerical value of the angle when expressed in radians. The capitalized function
6759:
6734:
6697:
6671:
6650:
6613:
6556:
6531:
6430:
6393:
6202:
6176:
6045:
6019:
5918:
5877:
5834:
5502:
5476:
5376:
5350:
5129:
3966:× 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under
3938:
3856:
766:
283:
749:
7144:
7030:
6818:
6738:
6701:
6654:
6646:
6617:
6609:
6535:
6527:
6478:
6434:
6426:
6397:
6389:
6291:
6049:
5964:
5781:
5749:
5696:
5615:
5506:
5380:
5372:
5154:
5133:
4934:
4884:, p. 151: "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
4669:
4474:
3876:
799:
is the numerical value of the radius of the pulley when expressed in centimeters and
6746:
Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022).
6579:
6544:
6206:
3117:
Some common angles, measured in radians. All the large polygons in this diagram are
6777:
6726:
6689:
6642:
6605:
6574:
6523:
6498:
6463:
6422:
6385:
6354:
6329:
6194:
6106:
6037:
5922:
5910:
5881:
5867:
5838:
5824:
5494:
5368:
5253:
5218:
5183:
5119:
4604:
3983:
3897:, the units of angular velocity and angular acceleration are s and s respectively.
3872:
878:
863:
812:
519:{\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}
6512:"The next 50 years of the SI: a review of the opportunities for the e-Science age"
6243:
Report of the 25th meeting to the International Committee for Weights and Measures
6225:
Report of the 21st meeting to the International Committee for Weights and Measures
5980:
A. Macfarlane (1893) "On the definitions of the trigonometric functions", page 9,
907:. With this change the formula for the angle subtended at the center of a circle,
6897:
4485:
to as the radian convention, which gives the simple formula for angular velocity
3995:
3118:
2852:
2629:
Conversely, to convert from degrees to radians, multiply magnitude in degrees by
286:
3621:
is the number of degrees, angle/degree, the number of radians, angle/radian, is
811:
the unit radian does not appear in the result. Similarly in the formula for the
7050:
6982:
6782:
6747:
6730:
6693:
6198:
6041:
5765:
5498:
4930:
4481:
3475:
is the numerical value of an angle when the angle is expressed in radians, i.e
1355:
194:
5124:
5099:
4911:
Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Revision 3
7159:
7111:
7035:
6972:
6962:
5993:
3949:
3913:
3480:
2320:. Thus, to convert from radians to degrees, multiply magnitude in radians by
928:
874:
847:
843:
765:
gives 1 SI radian as 1 m/m = 1. The key fact is that the SI radian is a
675:
6277:
6154:
6142:
5051:
4981:
4955:
4896:
4881:
4869:
891:
In particular, Quincey identifies Torrens' proposal to introduce a constant
7055:
7012:
6997:
6952:
6932:
6849:
5174:
French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)".
4685:
4637:
considered a proposal for making radians an SI base unit, using a constant
1345:
1294:
290:
6359:
6342:
5968:
5753:
4968:
309:
7175:
6927:
6882:
4540:
4466:
4226:
4006:
3953:
2891:
1328:
1247:
682:
362:
275:
160:
144:
4625:(SI) has long been a source of controversy and confusion." In 1960, the
4535:
first appeared in print on 5 June 1873, in examination questions set by
582:{\textstyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}
6368:
Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"".
5429:
4119:
2899:
457:
274:(SI) and is the standard unit of angular measure used in many areas of
6333:
6110:
5257:
5222:
5187:
4682:, a higher-dimensional analog of the radian which measures solid angle
4599:
in Chicago (acknowledged at page 167), and privately published in his
3214:
which is the basis of many other identities in mathematics, including
1331:'s unit system similarly considers angles to have an angle dimension.
6992:
6907:
6132:(in French), Consultative Committee for Units, May 1980, pp. 6–7
5914:
5872:
5829:
5542:
5417:
4679:
4630:
6502:
6468:
6443:
5890:
5847:
5804:
6872:
6764:
6676:
6405:
Torrens, A B (1 January 1986). "On Angles and Angular Quantities".
6181:
6024:
5939:
5481:
5355:
4434:
3130:
3126:
897:
equal to 1 inverse radian (1 rad) in a fashion similar to the
6561:
5730:(in Latin). Translated by Bruce, Ian. Definition 6, paragraph 316.
4657:
CCU Working Group on Angles and Dimensionless Quantities in the SI
353:
is the magnitude of the subtended angle in radians (= angle/rad),
7060:
7045:
6987:
6917:
6912:
6862:
6347:
Journal of Research of the National Bureau of Standards Section B
5201:
Oberhofer, E. S. (March 1992). "What happens to the 'radians'?".
4571:. In 1874, after a consultation with James Thomson, Muir adopted
4548:
3991:
3868:
2895:
1360:
227:
55:
6237:
6219:
5588:
Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi
4938:
4647:, but turned it down to avoid an upheaval to current practice.
4621:
As Paul Quincey et al. writes, "the status of angles within the
3113:
7075:
6887:
6867:
6857:
6799:
6278:
859:
289:, defined in the SI as 1 rad = 1 and expressed in terms of the
65:
61:
5054:, p. 151: "One radian corresponds to the angle for which
4984:, p. 151: "One radian corresponds to the angle for which
7090:
7085:
7070:
6947:
6942:
6937:
6922:
6877:
3999:
1293:
Current SI can be considered relative to this framework as a
293:
267:
104:
3204:{\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}
722:
is the (numerical value of the) subtended angle in radians,
7065:
7007:
7002:
6810:
6089:
Nelson, Robert A. (March 1984). "The supplementary units".
4010:
932:
4739:
3956:(mrad) is a thousandth of a radian (0.001 rad), i.e.
893:
705:
634:{\textstyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}
72:
of 1 radian. The circumference subtends an angle of 2
7040:
6967:
6957:
5940:"Earliest Known Uses of Some of the Words of Mathematics"
4014:
6745:
5548:
5435:
5423:
5315:
5310:
4969:"ISO 80000-3:2006 Quantities and Units - Space and Time"
4812:
4551:. He had used the term as early as 1871, while in 1869,
3941:) or newton-metre per radian (for torsional stiffness).
3916:, whilst if the phase angle difference of two waves is (
3129:
and most other branches of mathematics beyond practical
5742:
Plane Trigonometry: For the Use of Colleges and Schools
5235:
4845:. Vol. Part I: Plane Trigonometry. New York, USA:
3961:
94:
74:
6543:
Mohr, Peter J; Phillips, William D (1 February 2015).
5614:. Springer Science & Business Media. p. 134.
5450:
5448:
5446:
5444:
5407:
5405:
5392:
5390:
4579:
was not universally adopted for some time after this.
3461:{\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}
3420:
3359:
3337:{\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}
773:. In SI 2019, the SI radian is defined accordingly as
603:
595:
550:
532:
494:
466:
5634:
4775:
4714:
3871:
when angular measurements are required. For example,
3644:
3496:
3419:
3357:
3277:
3223:
3157:
3044:
2979:
2943:
2908:
2873:
2793:
2729:
2675:
2635:
2554:
2489:
2424:
2368:
2326:
2287:
1276:
1256:
1199:
945:
770:
402:
371:
322:
3934:
an integer, they are considered to be in antiphase.
6009:
5635:O'Connor, J. J.; Robertson, E. F. (February 2005).
5441:
5402:
5387:
4925:Protter, Murray H.; Morrey, Charles B. Jr. (1970),
4804:{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}
3879:(rad/s). One revolution per second corresponds to 2
1231:{\displaystyle x=\eta \theta =\theta /{\text{rad}}}
5769:
5611:The Mathematics of Measurement: A Critical History
4803:
4726:
4527:becoming widespread, the unit was commonly called
3840:
3598:
3460:
3405:
3336:
3262:
3203:
3092:
3029:
2962:
2929:
2882:
2841:
2777:
2709:
2658:
2618:
2539:
2474:
2404:
2351:
2312:
1282:
1262:
1230:
1185:
633:
581:
518:
423:
384:
341:
6155:International Bureau of Weights and Measures 2019
6143:International Bureau of Weights and Measures 2019
5052:International Bureau of Weights and Measures 2019
5010:, the radian can be defined in terms of the area
4982:International Bureau of Weights and Measures 2019
4956:International Bureau of Weights and Measures 2019
4897:International Bureau of Weights and Measures 2019
4882:International Bureau of Weights and Measures 2019
4870:International Bureau of Weights and Measures 2019
7157:
5673:(in Latin). Cambridge, England. pp. 94–95.
5562:"UnityDimensions—Wolfram Language Documentation"
4704:Other proposals include the abbreviation "rad" (
4002:beams is also usually measured in milliradians.
3159:
1290:if it is clear that the complete form is meant.
5963:, p. 6, London: Longmans, Green, and Co., 1890
4009:is an approximation of the milliradian used by
3890:is often radian per second per second (rad/s).
5719:Theoria Motus Corporum Solidorum seu Rigidorum
5150:Minds-on Physics: Advanced topics in mechanics
667:International Organization for Standardization
6826:
5724:Theory of the motion of solid or rigid bodies
4924:
4140: rad (around 4.8481 microradians).
6748:"On the dimension of angles and their units"
6542:
6479:"Dimensional angles and universal constants"
6476:
6279:International Bureau of Weights and Measures
5776:. Vol. 2. Dover Publications. pp.
5466:
5340:
5321:
5290:
5007:
4747:
4721:
4715:
3912:is an integer, they are considered to be in
3406:{\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}
3263:{\displaystyle {\frac {d}{dx}}\sin x=\cos x}
663:International Bureau of Weights and Measures
445: radians is equal to 360 degrees.
30:"㎭" redirects here. Not to be confused with
7117:Historical definitions of the SI base units
5594:]. Berlin: Akademie Verlag. p. 40.
5519:
4834:
4832:
6833:
6819:
6477:Lévy-Leblond, Jean-Marc (September 1998).
6441:
5336:
5334:
5332:
5330:
5286:
5086:Angular amplitude of swing No dimensions.
5084:. New Haven : Yale University Press.
4743:
3952:for submultiples are used with radians. A
54:
6781:
6763:
6675:
6578:
6560:
6467:
6358:
6311:
6180:
6084:
6082:
6023:
5871:
5828:
5592:Treatise on the Circumference of al-Kashi
5480:
5354:
5270:
5200:
5123:
4865:
4863:
4838:
4705:
4616:
4501:
2659:{\displaystyle {\pi }/{180}{\text{ rad}}}
1323:units library defines angle units with a
6343:"Angle as a fourth fundamental quantity"
5748:, Cambridge and London: MacMillan, 1864
5603:
5601:
5586:Luckey, Paul (1953) . Siggel, A. (ed.).
5520:Schabel, Matthias C.; Watanabe, Steven.
5077:
4829:
3944:
3112:
858:(N⋅m/rad), and not in the quantities of
839:American Association of Physics Teachers
6708:
6661:
6587:
6404:
6166:
5845:
5667:"Editoris notæ ad Harmoniam mensurarum"
5454:
5411:
5396:
5327:
5306:
5302:
5282:
5146:
5042:), in which case it has the units m⋅m."
5006:, p. 844: "Also, as alluded to in
5003:
4927:College Calculus with Analytic Geometry
4766:
4727:{\displaystyle \langle \theta \rangle }
4465:radian measure is normally credited to
691:
14:
7158:
6509:
6340:
6284:The International System of Units (SI)
6088:
6079:
5937:
5764:
5688:
5585:
5294:
5274:
5173:
5097:
4860:
4755:
4735:
4635:Consultative Committee for Units (CCU)
4441:as units, where one diameter part was
342:{\displaystyle \theta ={\frac {s}{r}}}
6814:
6624:
5664:
5607:
5598:
5298:
4154:
4151:
2963:{\displaystyle \pi /200{\text{ rad}}}
2352:{\displaystyle {180^{\circ }}/{\pi }}
2313:{\displaystyle {180^{\circ }}/{\pi }}
1835:
1829:
456:can be derived using the formula for
6510:Foster, Marcus P (1 December 2010).
6367:
5888:
5802:
5278:
4013:and other military organizations in
1263:{\displaystyle \operatorname {Sin} }
589:. This can be further simplified to
5891:"The Term "Radian" in Trigonometry"
5848:"The Term "Radian" in Trigonometry"
5805:"The Term "Radian" in Trigonometry"
5641:The MacTutor History of Mathematics
5608:Roche, John J. (21 December 1998).
4428:
3875:is typically expressed in the unit
2930:{\displaystyle 200^{\text{g}}/\pi }
2862:
424:{\displaystyle {\frac {2\pi r}{r}}}
308:One radian is defined as the angle
24:
6444:"Angles—Let's treat them squarely"
4796:
4778:
2281:As stated, one radian is equal to
2276:
1304:is assumed to hold, or similarly,
25:
7192:
6792:
6065:Le Système international d'unités
5772:History of Mathematical Notations
5715:
5692:Roger Cotes - Natural Philosopher
5078:Bridgman, Percy Williams (1922).
3413:, the evaluation of the integral
829:, radians appear in the units of
697:
385:{\displaystyle {\frac {\pi }{2}}}
7140:
7139:
6798:
6341:Romain, Jacques E. (July 1962).
6239:Consultative Committee for Units
6221:Consultative Committee for Units
5992:
4610:
6545:"Dimensionless units in the SI"
6442:Brownstein, K. R. (July 1997).
6307:from the original on 8 May 2021
6249:
6231:
6213:
6160:
6117:
6071:(in French), 1970, p. 12,
6056:
6003:
5986:
5974:
5953:
5931:
5794:
5758:
5734:
5709:
5689:Gowing, Ronald (27 June 2002).
5682:
5658:
5628:
5579:
5554:
5513:
5460:
5264:
5229:
5194:
5167:
5140:
5098:Prando, Giacomo (August 2020).
5091:
5071:
5045:
4997:
4698:
4559:, vacillated between the terms
1246:is the traditional function on
7127:System of units of measurement
5695:. Cambridge University Press.
4975:
4961:
4918:
4902:
4875:
4607:which is analogously defined.
4072:; for example Sweden used the
4021:. Each angular mil represents
3166:
3108:
1334:
1154:
1144:
1118:
1108:
1082:
1072:
834:"pedagogically unsatisfying".
702:Plane angle may be defined as
656:
13:
1:
5961:Longmans' School Trigonometry
5938:Miller, Jeff (Nov 23, 2009).
5153:. Kendall Hunt. p. 262.
4822:
4623:International System of Units
4581:Longmans' School Trigonometry
4147:SI multiples of radian (rad)
3886:Similarly, the unit used for
3867:The radian is widely used in
2894:, which is by definition 400
2686:magnitude of angle in degrees
2379:magnitude of angle in radians
899:introduction of the constant
750:the area of a circular sector
303:
272:International System of Units
6840:
5147:Leonard, William J. (1999).
4597:World's Columbian Exposition
2851:Radians can be converted to
1340:Conversion of common angles
641:. Multiplying both sides by
64:with the same length as the
7:
6483:American Journal of Physics
6448:American Journal of Physics
6314:American Journal of Physics
5537:Angles are treated as units
4663:
4617:§ Dimensional analysis
4502:§ Dimensional analysis
3988:(stadiametric) rangefinding
68:of that circle subtends an
10:
7197:
6647:10.1088/0026-1394/53/3/991
6610:10.1088/0026-1394/53/2/840
6528:10.1088/0026-1394/47/6/R01
6427:10.1088/0026-1394/22/1/002
6390:10.1088/0026-1394/18/1/002
5669:. In Smith, Robert (ed.).
5637:"Biography of Roger Cotes"
5522:"Boost.Units FAQ – 1.79.0"
5373:10.1088/0026-1394/53/3/998
4614:
4423:
3862:
2890:radians, which equals one
1316:in mathematical formulas.
1297:system where the equation
695:
36:
29:
7135:
7099:
7021:
6896:
6848:
6580:10.1088/0026-1394/52/1/40
6245:(Report). pp. 16–17.
6227:(Report). pp. 18–20.
6126:Report of the 7th meeting
5125:10.1038/s41567-020-0997-3
4929:(2nd ed.), Reading:
4437:(c. 1400) used so-called
4174:
4171:
4168:
4165:
4162:
4159:
4114:) and nanoradians (nrad,
223:
190:
156:
140:
136:
118:
110:
100:
88:
83:
53:
48:
7081:minute and second of arc
6783:10.1088/1681-7575/ac7bc2
6731:10.1088/1681-7575/abe0fc
6694:10.1088/1681-7575/ac023f
6241:(21–23 September 2021).
6199:10.1088/1681-7575/ab3fbf
6042:10.1088/1681-7575/ab27d7
5982:link at Internet Archive
5499:10.1088/1681-7575/aa7160
5322:Mohr & Phillips 2015
5008:Mohr & Phillips 2015
4691:
4675:Minute and second of arc
4601:Papers on Space Analysis
4587:when published in 1890.
4583:still called the radian
4557:University of St Andrews
3103:
917:, is modified to become
871:base unit of measurement
742:1 SI radian = 1 m/m
262:, denoted by the symbol
27:SI derived unit of angle
7122:2019 revision of the SI
5998:Geometry/Unified Angles
5846:Thomson, James (1910).
3142:trigonometric functions
1312:allows the omission of
39:Radian (disambiguation)
4847:Henry Holt and Company
4805:
4728:
4531:of an angle. The term
3842:
3600:
3462:
3407:
3338:
3264:
3205:
3122:
3094:
3031:
2964:
2931:
2884:
2843:
2779:
2711:
2660:
2620:
2541:
2476:
2406:
2353:
2314:
1284:
1264:
1232:
1187:
635:
583:
520:
425:
386:
343:
6360:10.6028/jres.066B.012
5665:Cotes, Roger (1722).
5566:reference.wolfram.com
4806:
4738:), and the constants
4729:
3945:Prefixes and variants
3843:
3601:
3463:
3408:
3350:differential equation
3339:
3265:
3206:
3116:
3095:
3032:
2965:
2932:
2885:
2883:{\displaystyle 2\pi }
2844:
2780:
2712:
2661:
2621:
2542:
2477:
2407:
2354:
2315:
1285:
1283:{\displaystyle \sin }
1265:
1233:
1188:
636:
584:
521:
426:
387:
344:
280:SI supplementary unit
7181:Units of plane angle
7023:Other accepted units
6807:at Wikimedia Commons
5889:Muir, Thos. (1910).
5803:Muir, Thos. (1910).
5081:Dimensional analysis
4773:
4712:
4592:Alexander Macfarlane
3986:manufacturers using
3895:dimensional analysis
3888:angular acceleration
3883:radians per second.
3853:exponential function
3642:
3494:
3417:
3355:
3275:
3221:
3155:
3042:
2977:
2941:
2906:
2871:
2791:
2727:
2673:
2633:
2552:
2487:
2422:
2366:
2324:
2285:
1274:
1254:
1197:
943:
852:angular acceleration
815:of a rolling wheel,
692:Dimensional analysis
593:
530:
464:
400:
369:
320:
37:For other uses, see
32:Rad (radiation unit)
7107:Conversion of units
6774:2022Metro..59e3001M
6723:2021Metro..58e2001L
6686:2021Metro..58e3002Q
6639:2016Metro..53..991M
6602:2016Metro..53..840Q
6571:2015Metro..52...40M
6495:1998AmJPh..66..814L
6460:1997AmJPh..65..605B
6419:1986Metro..22....1T
6382:1982Metro..18....1E
6326:1936AmJPh...4..175B
6223:(11–12 June 2013).
6191:2019Metro..56f5009K
6145:, pp. 174–175.
6103:1984PhTea..22..188N
6091:The Physics Teacher
6034:2019Metro..56d3001Q
5907:1910Natur..83..459M
5864:1910Natur..83..217T
5821:1910Natur..83..156M
5671:Harmonia mensurarum
5491:2017Metro..54..454Q
5365:2016Metro..53..998Q
5250:1993PhTea..31...84A
5238:The Physics Teacher
5215:1992PhTea..30..170O
5203:The Physics Teacher
5176:The Physics Teacher
5116:2020NatPh..16..888P
4471:Harmonia mensurarum
4148:
4037:of a circle and is
3893:For the purpose of
3855:(see, for example,
1341:
856:torsional stiffness
795:centimeters, where
726:is arc length, and
698:§ As a SI unit
357:is arc length, and
282:and is currently a
252: grad ≈ 63.662
131:... is equal to ...
84:General information
6900:with special names
5959:Frederick Sparks,
4971:. 17 January 2017.
4801:
4724:
4523:Prior to the term
4499:. As discussed in
4146:
4091:and the USSR used
3939:angular wavelength
3838:
3596:
3479:= angle /rad, the
3458:
3457:
3403:
3392:
3334:
3260:
3201:
3173:
3123:
3090:
3027:
2960:
2927:
2880:
2867:One revolution is
2839:
2775:
2707:
2656:
2616:
2537:
2472:
2402:
2349:
2310:
1339:
1280:
1260:
1228:
1183:
767:dimensionless unit
631:
629:
579:
573:
516:
510:
421:
382:
339:
233:
225:
200:
192:
166:
158:
150:
142:
7153:
7152:
7031:astronomical unit
6803:Media related to
6334:10.1119/1.1999110
6297:978-92-822-2272-0
6111:10.1119/1.2341516
5901:(2120): 459–460.
5740:Isaac Todhunter,
5716:Euler, Leonhard.
5702:978-0-521-52649-4
5621:978-0-387-91581-4
5291:Lévy-Leblond 1998
5258:10.1119/1.2343667
5223:10.1119/1.2343500
5188:10.1119/1.2343535
5160:978-0-7872-5412-4
5100:"A spectral unit"
4933:, p. APP-4,
4799:
4748:Lévy-Leblond 1998
4670:Angular frequency
4475:semicircumference
4421:
4420:
3908:) radians, where
3877:radian per second
3827:
3806:
3792:
3774:
3753:
3739:
3721:
3700:
3686:
3665:
3585:
3560:
3535:
3452:
3391:
3305:
3237:
3190:
3158:
3088:
3077:
3072:
3052:
3024:
3012:
3006:
2986:
2958:
2916:
2837:
2826:
2821:
2773:
2762:
2757:
2705:
2700:
2687:
2679:
2654:
2601:
2581:
2568:
2563:
2522:
2496:
2457:
2431:
2400:
2380:
2372:
2274:
2273:
1310:radian convention
1226:
1172:
1136:
1100:
1049:
1024:
999:
966:
628:
615:
572:
559:
509:
474:
419:
380:
337:
266:, is the unit of
256:
255:
119:Conversions
16:(Redirected from
7188:
7171:SI derived units
7143:
7142:
6835:
6828:
6821:
6812:
6811:
6802:
6787:
6785:
6767:
6742:
6705:
6679:
6658:
6621:
6584:
6582:
6564:
6539:
6506:
6473:
6471:
6438:
6401:
6364:
6362:
6337:
6308:
6306:
6290:(9th ed.),
6289:
6269:
6268:
6266:
6264:
6253:
6247:
6246:
6235:
6229:
6228:
6217:
6211:
6210:
6184:
6164:
6158:
6152:
6146:
6140:
6134:
6133:
6131:
6121:
6115:
6114:
6086:
6077:
6076:
6070:
6060:
6054:
6053:
6027:
6007:
6001:
5996:
5990:
5984:
5978:
5972:
5957:
5951:
5950:
5948:
5946:
5935:
5929:
5926:
5915:10.1038/083459d0
5885:
5875:
5873:10.1038/083217c0
5842:
5832:
5830:10.1038/083156a0
5798:
5792:
5791:
5775:
5762:
5756:
5738:
5732:
5731:
5729:
5713:
5707:
5706:
5686:
5680:
5679:
5662:
5656:
5655:
5653:
5652:
5643:. Archived from
5632:
5626:
5625:
5605:
5596:
5595:
5583:
5577:
5576:
5574:
5572:
5558:
5552:
5549:Mohr et al. 2022
5546:
5540:
5539:
5534:
5532:
5517:
5511:
5510:
5484:
5464:
5458:
5452:
5439:
5436:Mohr et al. 2022
5433:
5427:
5424:Mohr et al. 2022
5421:
5415:
5409:
5400:
5394:
5385:
5384:
5358:
5338:
5325:
5319:
5313:
5311:Mohr et al. 2022
5268:
5262:
5261:
5233:
5227:
5226:
5198:
5192:
5191:
5171:
5165:
5164:
5144:
5138:
5137:
5127:
5095:
5089:
5088:
5075:
5069:
5067:
5063:
5049:
5043:
5041:
5034:
5032:
5031:
5028:
5025:
5001:
4995:
4993:
4979:
4973:
4972:
4965:
4959:
4953:
4942:
4941:
4922:
4916:
4915:
4906:
4900:
4894:
4885:
4879:
4873:
4867:
4858:
4857:
4855:
4854:
4836:
4816:
4813:Mohr et al. 2022
4810:
4808:
4807:
4802:
4800:
4795:
4787:
4782:
4781:
4733:
4731:
4730:
4725:
4708:), the notation
4702:
4646:
4605:hyperbolic angle
4585:circular measure
4529:circular measure
4519:
4498:
4456:
4454:
4453:
4450:
4447:
4429:Pre-20th century
4149:
4145:
4139:
4137:
4136:
4133:
4130:
4129:
4117:
4113:
4106:
4104:
4103:
4100:
4097:
4087:
4085:
4084:
4081:
4078:
4071:
4069:
4068:
4067:
4062:
4059:
4052:
4050:
4049:
4046:
4043:
4036:
4034:
4033:
4030:
4027:
3984:telescopic sight
3981:
3979:
3978:
3975:
3972:
3964:
3959:
3930:) radians, with
3929:
3928:
3924:
3907:
3882:
3873:angular velocity
3847:
3845:
3844:
3839:
3828:
3826:
3818:
3817:
3808:
3804:
3803:
3802:
3797:
3793:
3785:
3775:
3773:
3765:
3764:
3755:
3751:
3750:
3749:
3744:
3740:
3732:
3722:
3720:
3712:
3711:
3702:
3698:
3697:
3696:
3691:
3687:
3679:
3666:
3658:
3634:
3629:
3616:
3605:
3603:
3602:
3597:
3586:
3584:
3576:
3575:
3566:
3561:
3559:
3551:
3550:
3541:
3536:
3534:
3526:
3525:
3516:
3467:
3465:
3464:
3459:
3453:
3451:
3450:
3449:
3433:
3425:
3412:
3410:
3409:
3404:
3393:
3390:
3389:
3388:
3375:
3371:
3370:
3360:
3343:
3341:
3340:
3335:
3306:
3304:
3303:
3302:
3289:
3288:
3279:
3269:
3267:
3266:
3261:
3238:
3236:
3225:
3210:
3208:
3207:
3202:
3191:
3186:
3175:
3172:
3119:regular polygons
3099:
3097:
3096:
3091:
3089:
3086:
3078:
3075:
3073:
3065:
3054:
3053:
3050:
3036:
3034:
3033:
3028:
3026:
3025:
3022:
3013:
3008:
3007:
3004:
2998:
2987:
2984:
2969:
2967:
2966:
2961:
2959:
2956:
2951:
2936:
2934:
2933:
2928:
2923:
2918:
2917:
2914:
2889:
2887:
2886:
2881:
2863:Between gradians
2858:
2848:
2846:
2845:
2840:
2838:
2835:
2827:
2824:
2822:
2814:
2803:
2802:
2784:
2782:
2781:
2776:
2774:
2771:
2763:
2760:
2758:
2750:
2739:
2738:
2716:
2714:
2713:
2708:
2706:
2703:
2701:
2693:
2688:
2685:
2680:
2678:angle in radians
2677:
2665:
2663:
2662:
2657:
2655:
2652:
2650:
2645:
2640:
2625:
2623:
2622:
2617:
2615:
2614:
2602:
2597:
2596:
2587:
2582:
2574:
2569:
2566:
2564:
2556:
2546:
2544:
2543:
2538:
2536:
2535:
2523:
2518:
2517:
2508:
2497:
2494:
2481:
2479:
2478:
2473:
2471:
2470:
2458:
2453:
2452:
2443:
2432:
2429:
2411:
2409:
2408:
2403:
2401:
2396:
2395:
2386:
2381:
2378:
2373:
2371:angle in degrees
2370:
2358:
2356:
2355:
2350:
2348:
2343:
2338:
2337:
2336:
2319:
2317:
2316:
2311:
2309:
2304:
2299:
2298:
2297:
2263:
2245:
2243:
2242:
2239:
2236:
2229:
2227:
2226:
2223:
2220:
2218:
2207:
2205:
2204:
2201:
2198:
2181:
2179:
2178:
2175:
2172:
2165:
2159:
2157:
2156:
2153:
2150:
2133:
2131:
2130:
2127:
2124:
2122:
2111:
2109:
2108:
2105:
2102:
2092:
2090:
2089:
2086:
2083:
2079:
2068:
2066:
2065:
2062:
2059:
2057:
2046:
2044:
2043:
2040:
2037:
2020:
2018:
2017:
2014:
2011:
2004:
2002:
2001:
1998:
1995:
1994:
1984:
1982:
1981:
1978:
1975:
1958:
1956:
1955:
1952:
1949:
1947:
1936:
1934:
1933:
1930:
1927:
1917:
1915:
1914:
1911:
1908:
1904:
1893:
1891:
1890:
1887:
1884:
1877:
1875:
1874:
1871:
1868:
1867:
1857:
1855:
1854:
1851:
1848:
1837:
1831:
1822:
1820:
1819:
1817:
1812:
1809:
1792:
1790:
1789:
1786:
1783:
1776:
1774:
1773:
1770:
1767:
1766:
1756:
1754:
1753:
1750:
1747:
1730:
1728:
1727:
1724:
1721:
1714:
1712:
1711:
1708:
1705:
1704:
1694:
1692:
1691:
1688:
1685:
1675:
1673:
1672:
1669:
1666:
1662:
1651:
1649:
1648:
1645:
1642:
1635:
1633:
1632:
1629:
1626:
1625:
1615:
1613:
1612:
1609:
1606:
1589:
1587:
1586:
1583:
1580:
1573:
1571:
1570:
1567:
1564:
1563:
1553:
1551:
1550:
1547:
1544:
1534:
1532:
1531:
1528:
1525:
1521:
1510:
1508:
1507:
1504:
1501:
1494:
1492:
1491:
1488:
1485:
1484:
1474:
1472:
1471:
1468:
1465:
1455:
1453:
1452:
1449:
1446:
1442:
1431:
1429:
1428:
1425:
1422:
1415:
1413:
1412:
1409:
1406:
1405:
1395:
1393:
1392:
1389:
1386:
1342:
1338:
1326:
1315:
1307:
1303:
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1245:
1241:
1237:
1235:
1234:
1229:
1227:
1224:
1222:
1192:
1190:
1189:
1184:
1173:
1171:
1163:
1162:
1161:
1142:
1137:
1135:
1127:
1126:
1125:
1106:
1101:
1099:
1091:
1090:
1089:
1070:
1050:
1048:
1040:
1039:
1030:
1025:
1023:
1015:
1014:
1005:
1000:
998:
990:
989:
980:
964:
938:
926:
916:
896:
887:
879:area of a circle
864:angular momentum
832:
828:
813:angular velocity
810:
806:
802:
798:
794:
780:
776:
764:
747:
743:
739:
729:
725:
721:
717:
687:
652:
644:
640:
638:
637:
632:
630:
627:
626:
617:
616:
613:
604:
588:
586:
585:
580:
578:
574:
571:
570:
561:
560:
557:
551:
525:
523:
522:
517:
515:
511:
508:
507:
495:
476:
475:
472:
455:
444:
437:
430:
428:
427:
422:
420:
415:
404:
391:
389:
388:
383:
381:
373:
360:
356:
352:
348:
346:
345:
340:
338:
330:
299:
251:
249:
248:
247:
243:
240:
234:
226:
218:
216:
215:
214:
210:
207:
201:
193:
185:
183:
182:
181:
176:
173:
167:
159:
151:
143:
77:
58:
46:
45:
21:
7196:
7195:
7191:
7190:
7189:
7187:
7186:
7185:
7156:
7155:
7154:
7149:
7131:
7112:Metric prefixes
7095:
7017:
6899:
6892:
6844:
6839:
6795:
6790:
6503:10.1119/1.18964
6469:10.1119/1.18616
6304:
6298:
6287:
6281:(20 May 2019),
6273:
6272:
6262:
6260:
6255:
6254:
6250:
6236:
6232:
6218:
6214:
6165:
6161:
6153:
6149:
6141:
6137:
6129:
6123:
6122:
6118:
6087:
6080:
6068:
6062:
6061:
6057:
6008:
6004:
5991:
5987:
5979:
5975:
5958:
5954:
5944:
5942:
5936:
5932:
5799:
5795:
5788:
5766:Cajori, Florian
5763:
5759:
5739:
5735:
5727:
5714:
5710:
5703:
5687:
5683:
5663:
5659:
5650:
5648:
5633:
5629:
5622:
5606:
5599:
5584:
5580:
5570:
5568:
5560:
5559:
5555:
5547:
5543:
5530:
5528:
5518:
5514:
5465:
5461:
5453:
5442:
5438:, pp. 8–9.
5434:
5430:
5422:
5418:
5410:
5403:
5395:
5388:
5349:(3): 998–1002.
5339:
5328:
5320:
5316:
5287:Brownstein 1997
5269:
5265:
5234:
5230:
5199:
5195:
5172:
5168:
5161:
5145:
5141:
5096:
5092:
5076:
5072:
5065:
5055:
5050:
5046:
5029:
5026:
5023:
5022:
5020:
5015:
5002:
4998:
4985:
4980:
4976:
4967:
4966:
4962:
4954:
4945:
4923:
4919:
4908:
4907:
4903:
4895:
4888:
4880:
4876:
4868:
4861:
4852:
4850:
4837:
4830:
4825:
4820:
4819:
4788:
4786:
4777:
4776:
4774:
4771:
4770:
4764:
4744:Brownstein 1997
4713:
4710:
4709:
4703:
4699:
4694:
4666:
4644:
4638:
4619:
4613:
4545:Queen's College
4506:
4486:
4461:The concept of
4451:
4448:
4445:
4444:
4442:
4431:
4426:
4143:
4134:
4131:
4127:
4126:
4125:
4123:
4115:
4111:
4101:
4098:
4095:
4094:
4092:
4082:
4079:
4076:
4075:
4073:
4065:
4063:
4060:
4057:
4056:
4054:
4047:
4044:
4041:
4040:
4038:
4031:
4028:
4025:
4024:
4022:
3976:
3973:
3970:
3969:
3967:
3962:
3958:1 rad = 10 mrad
3957:
3950:Metric prefixes
3947:
3926:
3922:
3917:
3905:
3880:
3865:
3857:Euler's formula
3819:
3813:
3809:
3807:
3798:
3784:
3780:
3779:
3766:
3760:
3756:
3754:
3745:
3731:
3727:
3726:
3713:
3707:
3703:
3701:
3692:
3678:
3674:
3673:
3657:
3643:
3640:
3639:
3627:
3622:
3614:
3577:
3571:
3567:
3565:
3552:
3546:
3542:
3540:
3527:
3521:
3517:
3515:
3495:
3492:
3491:
3445:
3441:
3434:
3426:
3424:
3418:
3415:
3414:
3384:
3380:
3376:
3366:
3362:
3361:
3358:
3356:
3353:
3352:
3298:
3294:
3290:
3284:
3280:
3278:
3276:
3273:
3272:
3229:
3224:
3222:
3219:
3218:
3176:
3174:
3162:
3156:
3153:
3152:
3111:
3106:
3085:
3074:
3064:
3049:
3045:
3043:
3040:
3039:
3021:
3017:
3003:
2999:
2997:
2983:
2978:
2975:
2974:
2970:. For example,
2955:
2947:
2942:
2939:
2938:
2919:
2913:
2909:
2907:
2904:
2903:
2872:
2869:
2868:
2865:
2856:
2834:
2823:
2813:
2798:
2794:
2792:
2789:
2788:
2770:
2759:
2749:
2734:
2730:
2728:
2725:
2724:
2702:
2692:
2684:
2676:
2674:
2671:
2670:
2651:
2646:
2641:
2636:
2634:
2631:
2630:
2610:
2606:
2592:
2588:
2586:
2573:
2565:
2555:
2553:
2550:
2549:
2531:
2527:
2513:
2509:
2507:
2493:
2488:
2485:
2484:
2466:
2462:
2448:
2444:
2442:
2428:
2423:
2420:
2419:
2391:
2387:
2385:
2377:
2369:
2367:
2364:
2363:
2344:
2339:
2332:
2328:
2327:
2325:
2322:
2321:
2305:
2300:
2293:
2289:
2288:
2286:
2283:
2282:
2279:
2277:Between degrees
2261:
2240:
2237:
2234:
2233:
2231:
2224:
2221:
2216:
2214:
2213:
2211:
2202:
2199:
2196:
2195:
2193:
2176:
2173:
2170:
2169:
2167:
2163:
2154:
2151:
2148:
2147:
2145:
2128:
2125:
2120:
2118:
2117:
2115:
2106:
2103:
2100:
2099:
2097:
2087:
2084:
2081:
2080:
2077:
2075:
2063:
2060:
2055:
2053:
2052:
2050:
2041:
2038:
2035:
2034:
2032:
2015:
2012:
2009:
2008:
2006:
1999:
1996:
1992:
1991:
1990:
1988:
1979:
1976:
1973:
1972:
1970:
1953:
1950:
1945:
1943:
1942:
1940:
1931:
1928:
1925:
1924:
1922:
1912:
1909:
1906:
1905:
1902:
1900:
1888:
1885:
1882:
1881:
1879:
1872:
1869:
1865:
1864:
1863:
1861:
1852:
1849:
1846:
1845:
1843:
1815:
1813:
1810:
1807:
1806:
1804:
1787:
1784:
1781:
1780:
1778:
1771:
1768:
1764:
1763:
1762:
1760:
1751:
1748:
1745:
1744:
1742:
1725:
1722:
1719:
1718:
1716:
1709:
1706:
1702:
1701:
1700:
1698:
1689:
1686:
1683:
1682:
1680:
1670:
1667:
1664:
1663:
1660:
1658:
1646:
1643:
1640:
1639:
1637:
1630:
1627:
1623:
1622:
1621:
1619:
1610:
1607:
1604:
1603:
1601:
1584:
1581:
1578:
1577:
1575:
1568:
1565:
1561:
1560:
1559:
1557:
1548:
1545:
1542:
1541:
1539:
1529:
1526:
1523:
1522:
1519:
1517:
1505:
1502:
1499:
1498:
1496:
1489:
1486:
1482:
1481:
1480:
1478:
1469:
1466:
1463:
1462:
1460:
1450:
1447:
1444:
1443:
1440:
1438:
1426:
1423:
1420:
1419:
1417:
1410:
1407:
1403:
1402:
1401:
1399:
1390:
1387:
1384:
1383:
1381:
1337:
1327:dimension, and
1324:
1313:
1305:
1298:
1275:
1272:
1271:
1270:can be denoted
1255:
1252:
1251:
1243:
1239:
1223:
1218:
1198:
1195:
1194:
1164:
1157:
1153:
1143:
1141:
1128:
1121:
1117:
1107:
1105:
1092:
1085:
1081:
1071:
1069:
1041:
1035:
1031:
1029:
1016:
1010:
1006:
1004:
991:
985:
981:
979:
944:
941:
940:
936:
918:
908:
905:
892:
882:
830:
816:
808:
804:
800:
796:
786:
778:
774:
752:
745:
741:
731:
727:
723:
719:
703:
700:
694:
685:
659:
646:
642:
622:
618:
612:
605:
602:
594:
591:
590:
566:
562:
556:
552:
549:
545:
531:
528:
527:
503:
499:
493:
489:
471:
467:
465:
462:
461:
449:
439:
432:
405:
403:
401:
398:
397:
372:
370:
367:
366:
358:
354:
350:
329:
321:
318:
317:
306:
297:
287:SI derived unit
245:
244:
241:
238:
237:
235:
232:
224:
212:
211:
208:
205:
204:
202:
199:
191:
179:
177:
174:
171:
170:
168:
165:
157:
149:
141:
79:
73:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7194:
7184:
7183:
7178:
7173:
7168:
7151:
7150:
7148:
7147:
7136:
7133:
7132:
7130:
7129:
7124:
7119:
7114:
7109:
7103:
7101:
7097:
7096:
7094:
7093:
7088:
7083:
7078:
7073:
7068:
7063:
7058:
7053:
7048:
7043:
7038:
7033:
7027:
7025:
7019:
7018:
7016:
7015:
7010:
7005:
7000:
6995:
6990:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6918:degree Celsius
6915:
6910:
6904:
6902:
6898:Derived units
6894:
6893:
6891:
6890:
6885:
6880:
6875:
6870:
6865:
6860:
6854:
6852:
6846:
6845:
6838:
6837:
6830:
6823:
6815:
6809:
6808:
6794:
6793:External links
6791:
6789:
6788:
6743:
6706:
6659:
6633:(3): 991–997.
6622:
6596:(2): 840–845.
6585:
6540:
6522:(6): R41–R51.
6507:
6489:(9): 814–815.
6474:
6454:(7): 605–614.
6439:
6402:
6365:
6338:
6320:(4): 175–179.
6309:
6296:
6274:
6271:
6270:
6248:
6230:
6212:
6159:
6157:, p. 179.
6147:
6135:
6116:
6097:(3): 188–193.
6078:
6055:
6002:
5985:
5973:
5971:(1891 edition)
5952:
5930:
5928:
5927:
5886:
5843:
5793:
5786:
5757:
5733:
5708:
5701:
5681:
5657:
5627:
5620:
5597:
5578:
5553:
5541:
5512:
5475:(4): 454–460.
5459:
5440:
5428:
5416:
5401:
5386:
5326:
5314:
5271:Brinsmade 1936
5263:
5228:
5209:(3): 170–171.
5193:
5182:(5): 260–261.
5166:
5159:
5139:
5104:Nature Physics
5090:
5070:
5044:
4996:
4974:
4960:
4958:, p. 151.
4943:
4931:Addison-Wesley
4917:
4901:
4899:, p. 137.
4886:
4874:
4859:
4827:
4826:
4824:
4821:
4818:
4817:
4798:
4794:
4791:
4785:
4780:
4762:
4723:
4720:
4717:
4706:Brinsmade 1936
4696:
4695:
4693:
4690:
4689:
4688:
4683:
4677:
4672:
4665:
4662:
4642:
4612:
4609:
4555:, then of the
4482:Leonhard Euler
4439:diameter parts
4430:
4427:
4425:
4422:
4419:
4418:
4415:
4412:
4409:
4406:
4403:
4399:
4398:
4395:
4392:
4389:
4386:
4383:
4379:
4378:
4375:
4372:
4369:
4366:
4363:
4359:
4358:
4355:
4352:
4349:
4346:
4343:
4339:
4338:
4335:
4332:
4329:
4326:
4323:
4319:
4318:
4315:
4312:
4309:
4306:
4303:
4299:
4298:
4295:
4292:
4289:
4286:
4283:
4279:
4278:
4275:
4272:
4269:
4266:
4263:
4259:
4258:
4255:
4252:
4249:
4246:
4243:
4239:
4238:
4235:
4232:
4229:
4224:
4221:
4217:
4216:
4213:
4210:
4207:
4204:
4201:
4197:
4196:
4193:
4190:
4187:
4184:
4181:
4177:
4176:
4173:
4170:
4167:
4164:
4161:
4157:
4156:
4153:
3946:
3943:
3864:
3861:
3849:
3848:
3837:
3834:
3831:
3825:
3822:
3816:
3812:
3801:
3796:
3791:
3788:
3783:
3778:
3772:
3769:
3763:
3759:
3748:
3743:
3738:
3735:
3730:
3725:
3719:
3716:
3710:
3706:
3695:
3690:
3685:
3682:
3677:
3672:
3669:
3664:
3661:
3656:
3653:
3650:
3647:
3607:
3606:
3595:
3592:
3589:
3583:
3580:
3574:
3570:
3564:
3558:
3555:
3549:
3545:
3539:
3533:
3530:
3524:
3520:
3514:
3511:
3508:
3505:
3502:
3499:
3456:
3448:
3444:
3440:
3437:
3432:
3429:
3423:
3402:
3399:
3396:
3387:
3383:
3379:
3374:
3369:
3365:
3345:
3344:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3301:
3297:
3293:
3287:
3283:
3270:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3235:
3232:
3228:
3212:
3211:
3200:
3197:
3194:
3189:
3185:
3182:
3179:
3171:
3168:
3165:
3161:
3110:
3107:
3105:
3102:
3101:
3100:
3084:
3081:
3071:
3068:
3063:
3060:
3057:
3048:
3037:
3020:
3016:
3011:
3002:
2996:
2993:
2990:
2982:
2954:
2950:
2946:
2926:
2922:
2912:
2879:
2876:
2864:
2861:
2833:
2830:
2820:
2817:
2812:
2809:
2806:
2801:
2797:
2786:
2785:
2769:
2766:
2756:
2753:
2748:
2745:
2742:
2737:
2733:
2718:
2717:
2699:
2696:
2691:
2683:
2649:
2644:
2639:
2627:
2626:
2613:
2609:
2605:
2600:
2595:
2591:
2585:
2580:
2577:
2572:
2562:
2559:
2547:
2534:
2530:
2526:
2521:
2516:
2512:
2506:
2503:
2500:
2492:
2482:
2469:
2465:
2461:
2456:
2451:
2447:
2441:
2438:
2435:
2427:
2413:
2412:
2399:
2394:
2390:
2384:
2376:
2347:
2342:
2335:
2331:
2308:
2303:
2296:
2292:
2278:
2275:
2272:
2271:
2268:
2265:
2258:
2254:
2253:
2250:
2247:
2209:
2190:
2189:
2186:
2183:
2161:
2142:
2141:
2138:
2135:
2113:
2094:
2093:
2073:
2070:
2048:
2029:
2028:
2025:
2022:
1986:
1967:
1966:
1963:
1960:
1938:
1919:
1918:
1898:
1895:
1859:
1840:
1839:
1833:
1827:
1824:
1801:
1800:
1797:
1794:
1758:
1739:
1738:
1735:
1732:
1696:
1677:
1676:
1656:
1653:
1617:
1598:
1597:
1594:
1591:
1555:
1536:
1535:
1515:
1512:
1476:
1457:
1456:
1436:
1433:
1397:
1378:
1377:
1374:
1371:
1368:
1364:
1363:
1358:
1353:
1348:
1336:
1333:
1279:
1259:
1221:
1217:
1214:
1211:
1208:
1205:
1202:
1182:
1179:
1176:
1170:
1167:
1160:
1156:
1152:
1149:
1146:
1140:
1134:
1131:
1124:
1120:
1116:
1113:
1110:
1104:
1098:
1095:
1088:
1084:
1080:
1077:
1074:
1068:
1065:
1062:
1059:
1056:
1053:
1047:
1044:
1038:
1034:
1028:
1022:
1019:
1013:
1009:
1003:
997:
994:
988:
984:
978:
975:
972:
969:
963:
960:
957:
954:
951:
948:
903:
744:= 1. However,
693:
690:
658:
655:
625:
621:
611:
608:
601:
598:
577:
569:
565:
555:
548:
544:
541:
538:
535:
514:
506:
502:
498:
492:
488:
485:
482:
479:
470:
418:
414:
411:
408:
379:
376:
336:
333:
328:
325:
305:
302:
254:
253:
230:
221:
220:
197:
188:
187:
163:
154:
153:
147:
138:
137:
134:
133:
128:
121:
120:
116:
115:
112:
108:
107:
102:
98:
97:
92:
86:
85:
81:
80:
59:
51:
50:
26:
9:
6:
4:
3:
2:
7193:
7182:
7179:
7177:
7174:
7172:
7169:
7167:
7166:Natural units
7164:
7163:
7161:
7146:
7138:
7137:
7134:
7128:
7125:
7123:
7120:
7118:
7115:
7113:
7110:
7108:
7105:
7104:
7102:
7098:
7092:
7089:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7069:
7067:
7064:
7062:
7059:
7057:
7054:
7052:
7051:degree of arc
7049:
7047:
7044:
7042:
7039:
7037:
7034:
7032:
7029:
7028:
7026:
7024:
7020:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6996:
6994:
6991:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6905:
6903:
6901:
6895:
6889:
6886:
6884:
6881:
6879:
6876:
6874:
6871:
6869:
6866:
6864:
6861:
6859:
6856:
6855:
6853:
6851:
6847:
6843:
6836:
6831:
6829:
6824:
6822:
6817:
6816:
6813:
6806:
6801:
6797:
6796:
6784:
6779:
6775:
6771:
6766:
6761:
6758:(5): 053001.
6757:
6753:
6749:
6744:
6740:
6736:
6732:
6728:
6724:
6720:
6717:(5): 052001.
6716:
6712:
6707:
6703:
6699:
6695:
6691:
6687:
6683:
6678:
6673:
6670:(5): 053002.
6669:
6665:
6660:
6656:
6652:
6648:
6644:
6640:
6636:
6632:
6628:
6623:
6619:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6586:
6581:
6576:
6572:
6568:
6563:
6558:
6554:
6550:
6546:
6541:
6537:
6533:
6529:
6525:
6521:
6517:
6513:
6508:
6504:
6500:
6496:
6492:
6488:
6484:
6480:
6475:
6470:
6465:
6461:
6457:
6453:
6449:
6445:
6440:
6436:
6432:
6428:
6424:
6420:
6416:
6412:
6408:
6403:
6399:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6366:
6361:
6356:
6352:
6348:
6344:
6339:
6335:
6331:
6327:
6323:
6319:
6315:
6310:
6303:
6299:
6293:
6286:
6285:
6280:
6276:
6275:
6258:
6252:
6244:
6240:
6234:
6226:
6222:
6216:
6208:
6204:
6200:
6196:
6192:
6188:
6183:
6178:
6175:(6): 065009.
6174:
6170:
6163:
6156:
6151:
6144:
6139:
6128:
6127:
6120:
6112:
6108:
6104:
6100:
6096:
6092:
6085:
6083:
6074:
6067:
6066:
6059:
6051:
6047:
6043:
6039:
6035:
6031:
6026:
6021:
6018:(4): 043001.
6017:
6013:
6006:
5999:
5995:
5989:
5983:
5977:
5970:
5966:
5962:
5956:
5941:
5934:
5924:
5920:
5916:
5912:
5908:
5904:
5900:
5896:
5892:
5887:
5883:
5879:
5874:
5869:
5865:
5861:
5858:(2112): 217.
5857:
5853:
5849:
5844:
5840:
5836:
5831:
5826:
5822:
5818:
5815:(2110): 156.
5814:
5810:
5806:
5801:
5800:
5797:
5789:
5787:0-486-67766-4
5783:
5779:
5774:
5773:
5767:
5761:
5755:
5751:
5747:
5743:
5737:
5725:
5721:
5720:
5712:
5704:
5698:
5694:
5693:
5685:
5677:
5672:
5668:
5661:
5647:on 2012-10-19
5646:
5642:
5638:
5631:
5623:
5617:
5613:
5612:
5604:
5602:
5593:
5589:
5582:
5567:
5563:
5557:
5550:
5545:
5538:
5527:
5526:www.boost.org
5523:
5516:
5508:
5504:
5500:
5496:
5492:
5488:
5483:
5478:
5474:
5470:
5463:
5456:
5451:
5449:
5447:
5445:
5437:
5432:
5425:
5420:
5413:
5408:
5406:
5398:
5393:
5391:
5382:
5378:
5374:
5370:
5366:
5362:
5357:
5352:
5348:
5344:
5337:
5335:
5333:
5331:
5323:
5318:
5312:
5308:
5304:
5300:
5296:
5292:
5288:
5284:
5280:
5276:
5272:
5267:
5259:
5255:
5251:
5247:
5243:
5239:
5232:
5224:
5220:
5216:
5212:
5208:
5204:
5197:
5189:
5185:
5181:
5177:
5170:
5162:
5156:
5152:
5151:
5143:
5135:
5131:
5126:
5121:
5117:
5113:
5109:
5105:
5101:
5094:
5087:
5083:
5082:
5074:
5062:
5058:
5053:
5048:
5040:
5037:
5018:
5014:of a sector (
5013:
5009:
5005:
5000:
4992:
4988:
4983:
4978:
4970:
4964:
4957:
4952:
4950:
4948:
4940:
4936:
4932:
4928:
4921:
4913:
4912:
4905:
4898:
4893:
4891:
4883:
4878:
4871:
4866:
4864:
4848:
4844:
4843:
4835:
4833:
4828:
4814:
4792:
4789:
4783:
4768:
4761:
4757:
4753:
4749:
4745:
4741:
4737:
4718:
4707:
4701:
4697:
4687:
4684:
4681:
4678:
4676:
4673:
4671:
4668:
4667:
4661:
4658:
4652:
4648:
4641:
4636:
4632:
4628:
4624:
4618:
4611:As an SI unit
4608:
4606:
4602:
4598:
4593:
4588:
4586:
4582:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4537:James Thomson
4534:
4530:
4526:
4521:
4517:
4513:
4509:
4504:
4503:
4497:
4493:
4489:
4483:
4478:
4476:
4472:
4468:
4464:
4459:
4440:
4436:
4417:quettaradian
4416:
4413:
4410:
4408:quectoradian
4407:
4404:
4401:
4400:
4396:
4393:
4390:
4387:
4384:
4381:
4380:
4376:
4373:
4370:
4367:
4364:
4361:
4360:
4356:
4353:
4350:
4347:
4344:
4341:
4340:
4336:
4333:
4330:
4327:
4324:
4321:
4320:
4316:
4313:
4310:
4307:
4304:
4301:
4300:
4296:
4293:
4290:
4287:
4284:
4281:
4280:
4276:
4273:
4270:
4267:
4264:
4261:
4260:
4256:
4253:
4250:
4247:
4244:
4241:
4240:
4236:
4233:
4230:
4228:
4225:
4222:
4219:
4218:
4214:
4211:
4208:
4205:
4202:
4199:
4198:
4194:
4191:
4188:
4185:
4182:
4179:
4178:
4158:
4152:Submultiples
4150:
4144:
4141:
4121:
4108:
4090:
4020:
4016:
4012:
4008:
4003:
4001:
3997:
3993:
3989:
3985:
3965:
3960:. There are 2
3955:
3951:
3942:
3940:
3935:
3933:
3920:
3915:
3911:
3903:
3898:
3896:
3891:
3889:
3884:
3878:
3874:
3870:
3860:
3858:
3854:
3835:
3832:
3829:
3823:
3820:
3814:
3810:
3799:
3794:
3789:
3786:
3781:
3776:
3770:
3767:
3761:
3757:
3746:
3741:
3736:
3733:
3728:
3723:
3717:
3714:
3708:
3704:
3693:
3688:
3683:
3680:
3675:
3670:
3667:
3662:
3659:
3654:
3651:
3648:
3645:
3638:
3637:
3636:
3632:
3625:
3620:
3612:
3593:
3590:
3587:
3581:
3578:
3572:
3568:
3562:
3556:
3553:
3547:
3543:
3537:
3531:
3528:
3522:
3518:
3512:
3509:
3506:
3503:
3500:
3497:
3490:
3489:
3488:
3486:
3483:for sin
3482:
3481:Taylor series
3478:
3474:
3469:
3454:
3446:
3442:
3438:
3435:
3430:
3427:
3421:
3400:
3397:
3394:
3385:
3381:
3377:
3372:
3367:
3363:
3351:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3299:
3295:
3291:
3285:
3281:
3271:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3233:
3230:
3226:
3217:
3216:
3215:
3198:
3195:
3192:
3187:
3183:
3180:
3177:
3169:
3163:
3151:
3150:
3149:
3147:
3143:
3139:
3134:
3132:
3128:
3120:
3115:
3082:
3079:
3069:
3066:
3061:
3058:
3055:
3046:
3038:
3018:
3014:
3009:
3000:
2994:
2991:
2988:
2980:
2973:
2972:
2971:
2952:
2948:
2944:
2924:
2920:
2910:
2901:
2897:
2893:
2877:
2874:
2860:
2854:
2849:
2831:
2828:
2818:
2815:
2810:
2807:
2804:
2799:
2795:
2767:
2764:
2754:
2751:
2746:
2743:
2740:
2735:
2731:
2723:
2722:
2721:
2720:For example:
2697:
2694:
2689:
2681:
2669:
2668:
2667:
2647:
2642:
2637:
2611:
2607:
2603:
2598:
2593:
2589:
2583:
2578:
2575:
2570:
2560:
2557:
2548:
2532:
2528:
2524:
2519:
2514:
2510:
2504:
2501:
2498:
2490:
2483:
2467:
2463:
2459:
2454:
2449:
2445:
2439:
2436:
2433:
2425:
2418:
2417:
2416:
2415:For example:
2397:
2392:
2388:
2382:
2374:
2362:
2361:
2360:
2345:
2340:
2333:
2329:
2306:
2301:
2294:
2290:
2269:
2266:
2259:
2256:
2255:
2251:
2248:
2210:
2192:
2191:
2187:
2184:
2162:
2144:
2143:
2139:
2136:
2114:
2096:
2095:
2074:
2071:
2049:
2031:
2030:
2026:
2023:
1987:
1969:
1968:
1964:
1961:
1939:
1921:
1920:
1899:
1896:
1860:
1842:
1841:
1834:
1828:
1825:
1803:
1802:
1798:
1795:
1759:
1741:
1740:
1736:
1733:
1697:
1679:
1678:
1657:
1654:
1618:
1600:
1599:
1595:
1592:
1556:
1538:
1537:
1516:
1513:
1477:
1459:
1458:
1437:
1434:
1398:
1380:
1379:
1375:
1372:
1369:
1366:
1365:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1343:
1332:
1330:
1322:
1317:
1311:
1301:
1296:
1291:
1277:
1257:
1249:
1219:
1215:
1212:
1209:
1206:
1203:
1200:
1180:
1177:
1174:
1168:
1165:
1158:
1150:
1147:
1138:
1132:
1129:
1122:
1114:
1111:
1102:
1096:
1093:
1086:
1078:
1075:
1066:
1063:
1060:
1057:
1054:
1051:
1045:
1042:
1036:
1032:
1026:
1020:
1017:
1011:
1007:
1001:
995:
992:
986:
982:
976:
973:
970:
967:
961:
958:
955:
952:
949:
946:
934:
930:
929:Taylor series
925:
921:
915:
911:
906:
902:
895:
889:
886:
880:
876:
875:base quantity
872:
867:
865:
861:
857:
854:(rad/s), and
853:
849:
848:angular speed
845:
844:angle measure
840:
835:
827:
823:
819:
814:
793:
789:
782:
772:
768:
763:
759:
755:
751:
738:
734:
716:
712:
708:
707:
699:
689:
684:
679:
677:
676:degree symbol
672:
668:
664:
654:
650:
623:
619:
609:
606:
599:
596:
575:
567:
563:
553:
546:
542:
539:
536:
533:
512:
504:
500:
496:
490:
486:
483:
480:
477:
468:
459:
453:
448:The relation
446:
443:
436:
416:
412:
409:
406:
393:
377:
374:
364:
361:is radius. A
334:
331:
326:
323:
315:
311:
301:
295:
292:
288:
285:
284:dimensionless
281:
277:
273:
269:
265:
261:
231:
229:
222:
198:
196:
189:
164:
162:
155:
148:
146:
139:
135:
132:
129:
127:
123:
122:
117:
113:
109:
106:
103:
99:
96:
93:
91:
87:
82:
76:
71:
67:
63:
57:
52:
47:
44:
40:
33:
19:
7056:electronvolt
6977:
6755:
6751:
6714:
6710:
6667:
6663:
6630:
6626:
6593:
6589:
6555:(1): 40–47.
6552:
6548:
6519:
6515:
6486:
6482:
6451:
6447:
6410:
6406:
6373:
6369:
6350:
6346:
6317:
6313:
6283:
6261:. Retrieved
6251:
6233:
6215:
6172:
6168:
6162:
6150:
6138:
6125:
6119:
6094:
6090:
6072:
6064:
6058:
6015:
6011:
6005:
6000:at Wikibooks
5988:
5976:
5960:
5955:
5943:. Retrieved
5933:
5898:
5894:
5855:
5851:
5812:
5808:
5796:
5771:
5760:
5741:
5736:
5723:
5718:
5711:
5691:
5684:
5674:
5670:
5660:
5649:. Retrieved
5645:the original
5640:
5630:
5610:
5591:
5587:
5581:
5569:. Retrieved
5565:
5556:
5551:, p. 3.
5544:
5536:
5529:. Retrieved
5525:
5515:
5472:
5468:
5462:
5455:Quincey 2021
5431:
5426:, p. 6.
5419:
5412:Torrens 1986
5397:Quincey 2016
5346:
5342:
5317:
5307:Leonard 2021
5303:Quincey 2021
5283:Torrens 1986
5266:
5244:(2): 84–87.
5241:
5237:
5231:
5206:
5202:
5196:
5179:
5175:
5169:
5149:
5142:
5107:
5103:
5093:
5085:
5080:
5073:
5060:
5056:
5047:
5038:
5035:
5016:
5011:
5004:Quincey 2016
4999:
4990:
4986:
4977:
4963:
4926:
4920:
4910:
4904:
4877:
4851:. Retrieved
4842:Trigonometry
4841:
4767:Quincey 2021
4759:
4751:
4700:
4686:Trigonometry
4656:
4653:
4649:
4639:
4620:
4600:
4589:
4584:
4580:
4576:
4572:
4568:
4564:
4560:
4539:(brother of
4532:
4528:
4524:
4522:
4515:
4511:
4507:
4500:
4495:
4491:
4487:
4479:
4470:
4462:
4460:
4438:
4432:
4411:10 rad
4402:10 rad
4397:ronnaradian
4391:10 rad
4388:rontoradian
4382:10 rad
4377:yottaradian
4371:10 rad
4368:yoctoradian
4362:10 rad
4357:zettaradian
4351:10 rad
4348:zeptoradian
4342:10 rad
4331:10 rad
4322:10 rad
4311:10 rad
4308:femtoradian
4302:10 rad
4291:10 rad
4282:10 rad
4271:10 rad
4262:10 rad
4251:10 rad
4248:microradian
4242:10 rad
4231:10 rad
4220:10 rad
4215:hectoradian
4209:10 rad
4206:centiradian
4200:10 rad
4189:10 rad
4180:10 rad
4142:
4109:
4088:
4004:
3948:
3936:
3931:
3918:
3909:
3901:
3899:
3892:
3885:
3866:
3850:
3630:
3623:
3618:
3610:
3608:
3484:
3476:
3472:
3470:
3346:
3213:
3135:
3124:
2866:
2850:
2787:
2719:
2628:
2414:
2280:
1350:
1318:
1309:
1299:
1295:natural unit
1292:
1248:pure numbers
935:of an angle
923:
919:
913:
909:
900:
890:
884:
868:
837:In 1993 the
836:
825:
821:
817:
791:
787:
783:
761:
757:
753:
736:
732:
714:
710:
704:
701:
680:
670:
660:
648:
451:
447:
441:
434:
394:
307:
291:SI base unit
263:
259:
257:
145:milliradians
130:
125:
101:Unit of
69:
60:An arc of a
43:
6376:(1): 1–12.
5295:Foster 2010
5275:Romain 1962
4756:Foster 2010
4736:Romain 1962
4575:. The name
4553:Thomas Muir
4541:Lord Kelvin
4467:Roger Cotes
4328:attoradian
4317:petaradian
4297:teraradian
4288:picoradian
4277:gigaradian
4268:nanoradian
4257:megaradian
4237:kiloradian
4227:milliradian
4195:decaradian
4186:deciradian
4122:, which is
4116:10 rad
4112:10 rad
4007:angular mil
3954:milliradian
3136:Results in
3109:Mathematics
2123:or 2𝜏 or α
1335:Conversions
1329:Mathematica
1325:plane_angle
683:degree sign
657:Unit symbol
365:is exactly
363:right angle
276:mathematics
219:° ≈ 57.296°
124:1 rad
114:rad,
90:Unit system
7160:Categories
6850:Base units
6765:2203.12392
6752:Metrologia
6711:Metrologia
6677:2108.05704
6664:Metrologia
6627:Metrologia
6590:Metrologia
6549:Metrologia
6516:Metrologia
6413:(1): 1–7.
6407:Metrologia
6370:Metrologia
6182:1810.12057
6169:Metrologia
6025:1909.08389
6012:Metrologia
5651:2006-04-21
5482:1705.03765
5469:Metrologia
5356:1604.02373
5343:Metrologia
5299:Mills 2016
5110:(8): 888.
4853:2017-08-12
4823:References
4615:See also:
4337:exaradian
4172:SI symbol
4163:SI symbol
4155:Multiples
4120:arc second
3996:divergence
3140:involving
2076:133
927:, and the
866:(kg⋅m/s).
862:(N⋅m) and
696:See also:
458:arc length
454:rad = 360°
304:Definition
6993:steradian
6908:becquerel
6739:234036217
6702:236547235
6655:126032642
6618:125438811
6562:1409.2794
6536:117711734
6435:250801509
6398:250750831
6353:(3): 97.
6050:198428043
5969:877238863
5754:500022958
5507:119418270
5381:119294905
5279:Eder 1982
5134:225445454
5066:1 rad = 1
4797:Θ
4793:π
4722:⟩
4719:θ
4716:⟨
4680:Steradian
4631:steradian
4480:In 1765,
4019:targeting
3833:⋯
3787:π
3777:−
3734:π
3681:π
3671:−
3660:π
3649:
3617:/180: if
3591:⋯
3563:−
3513:−
3501:
3487:becomes:
3422:∫
3398:−
3326:
3320:−
3311:
3255:
3243:
3181:
3167:→
3087: rad
3080:≈
3076: rad
3067:π
3062:⋅
3015:≈
3010:π
2995:⋅
2985: rad
2957: rad
2945:π
2925:π
2878:π
2836: rad
2829:≈
2825: rad
2816:π
2811:⋅
2800:∘
2772: rad
2765:≈
2761: rad
2752:π
2747:⋅
2736:∘
2704: rad
2695:π
2690:⋅
2653: rad
2638:π
2612:∘
2599:π
2594:∘
2584:⋅
2576:π
2567: rad
2558:π
2533:∘
2525:≈
2520:π
2515:∘
2505:⋅
2495: rad
2468:∘
2460:≈
2455:π
2450:∘
2440:⋅
2430: rad
2398:π
2393:∘
2383:⋅
2346:π
2334:∘
2307:π
2295:∘
1901:66
1659:33
1518:16
1306:1 rad = 1
1216:θ
1210:θ
1207:η
1178:⋯
1151:θ
1148:η
1139:−
1115:θ
1112:η
1079:θ
1076:η
1067:−
1064:θ
1061:η
1055:⋯
1027:−
977:−
962:
953:θ
950:
939:becomes:
850:(rad/s),
775:1 rad = 1
769:equal to
688:is used.
624:∘
614: rad
610:π
568:∘
558: rad
543:π
505:∘
497:θ
484:π
469:ℓ
410:π
392:radians.
375:π
324:θ
314:magnitude
310:subtended
298:rad = m/m
152:1000 mrad
7145:Category
7100:See also
6873:kilogram
6842:SI units
6302:archived
6207:53627142
5768:(1929).
4939:76087042
4664:See also
4590:In 1893
4435:al-Kashi
3992:reticles
3148:formula
3138:analysis
3131:geometry
3127:calculus
2896:gradians
2529:143.2394
1439:5
1361:Gradians
931:for the
740:, hence
718:, where
669:specify
647:360° = 2
438:. Thus,
349:, where
228:gradians
78:radians.
7061:hectare
7046:decibel
6988:sievert
6983:siemens
6913:coulomb
6863:candela
6770:Bibcode
6719:Bibcode
6682:Bibcode
6635:Bibcode
6598:Bibcode
6567:Bibcode
6491:Bibcode
6456:Bibcode
6415:Bibcode
6378:Bibcode
6322:Bibcode
6263:26 June
6187:Bibcode
6099:Bibcode
6030:Bibcode
5945:Sep 30,
5923:3971449
5903:Bibcode
5882:3980250
5860:Bibcode
5839:3958702
5817:Bibcode
5778:147–148
5487:Bibcode
5361:Bibcode
5246:Bibcode
5211:Bibcode
5112:Bibcode
5064:, thus
5033:
5021:
4769:), and
4645:= 1 rad
4549:Belfast
4455:
4443:
4424:History
4138:
4135:648,000
4124:
4105:
4093:
4086:
4074:
4070:
4055:
4051:
4039:
4035:
4023:
4015:gunnery
3980:
3968:
3869:physics
3863:Physics
3019:76.3944
2464:57.2958
2260:𝜏 or 2
2257:1 turn
2244:
2232:
2228:
2212:
2206:
2194:
2180:
2168:
2158:
2146:
2132:
2116:
2110:
2098:
2091:
2067:
2051:
2045:
2033:
2019:
2007:
2003:
1989:
1983:
1971:
1957:
1941:
1935:
1923:
1916:
1892:
1880:
1876:
1862:
1856:
1844:
1836:approx.
1830:approx.
1821:
1805:
1791:
1779:
1775:
1761:
1755:
1743:
1729:
1717:
1713:
1699:
1693:
1681:
1674:
1650:
1638:
1634:
1620:
1614:
1602:
1588:
1576:
1572:
1558:
1552:
1540:
1533:
1509:
1497:
1493:
1479:
1473:
1461:
1454:
1430:
1418:
1414:
1400:
1394:
1382:
1367:0 turn
1356:Degrees
1351:Radians
1308:. This
846:(rad),
779:rad = 1
296:(m) as
270:in the
250:
236:
217:
203:
195:degrees
184:
169:
18:Radians
7076:minute
7036:dalton
6978:radian
6973:pascal
6963:newton
6888:second
6868:kelvin
6858:ampere
6805:Radian
6737:
6700:
6653:
6616:
6534:
6433:
6396:
6294:
6259:. BIPM
6205:
6048:
5967:
5921:
5895:Nature
5880:
5852:Nature
5837:
5809:Nature
5784:
5752:
5726:]
5699:
5618:
5571:1 July
5505:
5379:
5157:
5132:
4937:
4746:), ◁ (
4577:radian
4573:radian
4569:radian
4567:, and
4565:radial
4533:radian
4525:radian
4192:darad
4169:Value
4160:Value
4089:streck
3994:. The
3805:
3752:
3699:
3083:0.7854
2832:0.4014
2768:0.0175
1832:57.3°
1826:1 rad
1593:22.5°
1370:0 rad
1193:where
965:
873:for a
860:torque
645:gives
260:radian
126:in ...
111:Symbol
75:π
66:radius
62:circle
49:Radian
7091:tonne
7086:neper
7071:litre
7013:weber
6998:tesla
6953:lumen
6948:katal
6943:joule
6938:hertz
6933:henry
6923:farad
6878:metre
6760:arXiv
6735:S2CID
6698:S2CID
6672:arXiv
6651:S2CID
6614:S2CID
6557:arXiv
6532:S2CID
6431:S2CID
6394:S2CID
6305:(PDF)
6288:(PDF)
6203:S2CID
6177:arXiv
6130:(PDF)
6069:(PDF)
6046:S2CID
6020:arXiv
5919:S2CID
5878:S2CID
5835:S2CID
5746:p. 10
5728:(PDF)
5722:[
5590:[
5531:5 May
5503:S2CID
5477:arXiv
5377:S2CID
5351:arXiv
5130:S2CID
4692:Notes
4543:) at
4414:Qrad
4405:qrad
4394:Rrad
4385:rrad
4374:Yrad
4365:yrad
4354:Zrad
4345:zrad
4334:Erad
4325:arad
4314:Prad
4305:frad
4294:Trad
4285:prad
4274:Grad
4265:nrad
4254:Mrad
4245:μrad
4234:krad
4223:mrad
4212:hrad
4203:crad
4183:drad
4175:Name
4166:Name
4000:laser
3914:phase
3635:, so
3633:/ 180
3146:limit
3104:Usage
2898:(400
2853:turns
2267:360°
2249:270°
2208:turn
2185:180°
2160:turn
2137:144°
2112:turn
2072:120°
2058:or 𝜏
2047:turn
1985:turn
1948:or 𝜏
1937:turn
1858:turn
1838:63.7
1823:turn
1818:or 𝜏
1757:turn
1695:turn
1616:turn
1554:turn
1475:turn
1396:turn
1346:Turns
1321:Boost
431:, or
294:metre
268:angle
161:turns
105:angle
70:angle
7066:hour
7008:watt
7003:volt
6928:gray
6883:mole
6292:ISBN
6265:2022
5965:OCLC
5947:2011
5782:ISBN
5750:OCLC
5697:ISBN
5616:ISBN
5573:2022
5533:2022
5155:ISBN
4935:LCCN
4627:CGPM
4102:6000
4083:6300
4064:2000
4032:6400
4017:and
4011:NATO
4005:The
3977:6283
2900:gons
2892:turn
2270:400
2264:rad
2252:300
2246:rad
2219:or ρ
2188:200
2182:rad
2140:160
2134:rad
2069:rad
2027:100
2024:90°
2021:rad
1962:72°
1959:rad
1897:60°
1894:rad
1796:45°
1793:rad
1734:36°
1731:rad
1655:30°
1652:rad
1590:rad
1514:15°
1511:rad
1432:rad
933:sine
665:and
661:The
643:360°
258:The
186:turn
7041:day
6968:ohm
6958:lux
6778:doi
6727:doi
6690:doi
6643:doi
6606:doi
6575:doi
6524:doi
6499:doi
6464:doi
6423:doi
6386:doi
6355:doi
6351:66B
6330:doi
6195:doi
6107:doi
6038:doi
5911:doi
5868:doi
5825:doi
5495:doi
5369:doi
5254:doi
5219:doi
5184:doi
5120:doi
4758:),
4750:),
4561:rad
4463:the
3998:of
3990:in
3790:180
3737:180
3684:180
3663:180
3646:sin
3609:If
3498:sin
3323:sin
3308:sin
3252:cos
3240:sin
3178:sin
3160:lim
3125:In
3070:200
3001:200
2992:1.2
2981:1.2
2953:200
2911:200
2819:180
2755:180
2698:180
2648:180
2590:180
2511:180
2502:2.5
2491:2.5
2446:180
2389:180
2330:180
2291:180
2235:3𝜏
2230:or
2166:or
2005:or
1965:80
1878:or
1799:50
1777:or
1737:40
1715:or
1636:or
1596:25
1574:or
1495:or
1435:5°
1416:or
1373:0°
1302:= 1
1278:sin
1258:Sin
1244:sin
1240:Sin
1225:rad
959:sin
947:Sin
924:ηrθ
807:by
781:.
756:= 2
746:rad
671:rad
651:rad
620:360
564:360
501:360
473:arc
264:rad
239:200
206:180
7176:Pi
7162::
6776:.
6768:.
6756:59
6754:.
6750:.
6733:.
6725:.
6715:58
6713:.
6696:.
6688:.
6680:.
6668:58
6666:.
6649:.
6641:.
6631:53
6629:.
6612:.
6604:.
6594:53
6592:.
6573:.
6565:.
6553:52
6551:.
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