5674:[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.]
4647:
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".
5672:
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.
5990:
7137:
3114:
56:
6796:
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.
3842:
942:
4629:. 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
5671:
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
4655:
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
4480:
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
4590:
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
4469:. 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
3637:
4646:
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
4049:% 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
3855:) can be elegantly stated when the functions' arguments are angles expressed in radians (and messy otherwise). More generally, in complex-number theory, the arguments of these functions are (dimensionless, possibly complex) numbers—without any reference to physical angles at all.
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:
4650:
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
2715:
2410:
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
3600:
2783:
2480:
3837:{\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
4453:
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.
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:
4805:
1236:
3411:
3268:
2664:
4728:
347:
2968:
2551:
2357:
2318:
1268:
2935:
432:
4835:
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:
2672:
2365:
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:
3489:
2726:
3896:
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
6069:
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.
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.
4501:, the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for example
6705:
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.
4103:. 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:
4868:, p. 151: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units."
6297:
666:
5632:
529:
6120:
662:
7112:
6238:
6059:
4656:"review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.
4622:
5517:
6006:
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:
6220:
4114:) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is the
3847:
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
6291:
5696:
5615:
5154:
1242:
is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while
5232:
Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".
3416:
3274:
6163:
Kalinin, Mikhail I (1 December 2019). "On the status of plane and solid angles in the
International System of Units (SI)".
4768:
4465:, who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book,
1196:
6584:
Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units".
6828:
5337:
Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI".
5781:
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
7018:
6234:
6216:
6071:[For some units of the SI, the CGPM still hasn't yet decided whether they are base units or derived units.]
4630:
3354:
3220:
4592:
7122:
5713:
3933:
A unit of reciprocal radian or inverse radian (rad) is involved in derived units such as meter per radian (for
2632:
748:
is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using
89:
6308:
Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their
Pedagogic Value When Introduced Explicitly".
5993:
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,
7176:
6837:
6278:
4707:
4618:
4540:
4429:
The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example,
271:
6507:
319:
5977:
5075:
2940:
2323:
2284:
4836:
3978:
of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by
1253:
4548:
2905:
2710:{\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180}}{\text{ rad}}}
402:
399:, expressed as an angle in radians, is the length of the circumference divided by the radius, which is
368:
4845:/ Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 73
7166:
3983:
2405:{\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}
7076:
5640:
5463:
Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems".
4670:
4552:
4532:
870:
5557:
4473:, 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
7117:
6800:
6658:
Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem".
5662:
4106:
Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad,
4014:
3093:{\displaystyle 50^{\text{g}}=50\cdot {\frac {\pi }{200}}{\text{ rad}}\approx 0.7854{\text{ rad}}}
1320:
313:
38:
17:
5773:
5766:
7161:
6821:
6132:
4910:. National Aeronautics and Space Administration, Goddard Space Flight Center. 2002. p. 12.
4842:
3141:
3137:
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
5605:
5144:
6252:
5741:
5686:
4947:
4945:
4943:
4905:
4888:
4886:
3595:{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}
3349:
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
6765:
6714:
6677:
6630:
6593:
6562:
6486:
6451:
6410:
6373:
6317:
6182:
6144:
6094:
6025:
5898:
5855:
5812:
5482:
5356:
5241:
5206:
5107:
4587:
3890:
3883:
3848:
2870:
1273:
851:
31:
4940:
4883:
4625:
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:
7102:
6253:"CCU Task Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB)"
3145:
396:
6769:
6718:
6681:
6634:
6621:
Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle".
6597:
6566:
6490:
6455:
6414:
6377:
6321:
6186:
6098:
6029:
5902:
5859:
5816:
5486:
5360:
5245:
5210:
5111:
4599:(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
6755:
6730:
6693:
6667:
6646:
6609:
6552:
6527:
6426:
6389:
6198:
6172:
6041:
6015:
5914:
5873:
5830:
5498:
5472:
5372:
5346:
5125:
3962:× 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under
3934:
3852:
766:
283:
749:
7140:
7026:
6814:
6734:
6697:
6650:
6642:
6613:
6605:
6531:
6523:
6474:
6430:
6422:
6393:
6385:
6287:
6045:
5960:
5777:
5745:
5692:
5611:
5502:
5376:
5368:
5150:
5129:
4930:
4880:, p. 151: "One radian corresponds to the angle for which s = r, thus 1 rad = 1."
4665:
4470:
3872:
6742:
Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022).
6575:
6540:
6202:
3117:
Some common angles, measured in radians. All the large polygons in this diagram are
6773:
6722:
6685:
6638:
6601:
6570:
6519:
6494:
6459:
6418:
6381:
6350:
6325:
6190:
6102:
6033:
5918:
5906:
5877:
5863:
5834:
5820:
5490:
5364:
5249:
5214:
5179:
5115:
4600:
3979:
3893:, the units of angular velocity and angular acceleration are s and s respectively.
3868:
878:
863:
812:
519:{\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}
6508:"The next 50 years of the SI: a review of the opportunities for the e-Science age"
6239:
Report of the 25th meeting to the International Committee for Weights and Measures
6221:
Report of the 21st meeting to the International Committee for Weights and Measures
5976:
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,
6893:
4481:
to as the radian convention, which gives the simple formula for angular velocity
3991:
3118:
2852:
730:
is radius. One SI radian corresponds to the angle expressed in radians for which
286:
811:
the unit radian does not appear in the result. Similarly in the formula for the
7046:
6978:
6778:
6743:
6726:
6689:
6194:
6037:
5761:
5494:
4926:
4477:
1355:
194:
5120:
5095:
4907:
Ocean Optics Protocols for Satellite Ocean Color Sensor Validation, Revision 3
7155:
7107:
7031:
6968:
6958:
5989:
3945:
3909:
3476:
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:
6273:
6150:
6138:
5047:
4977:
4951:
4892:
4877:
4865:
891:
In particular, Quincey identifies Torrens' proposal to introduce a constant
7051:
7008:
6993:
6948:
6928:
6845:
5170:
French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)".
4681:
4633:
considered a proposal for making radians an SI base unit, using a constant
1345:
1294:
290:
6355:
6338:
5964:
5749:
4964:
309:
7171:
6923:
6878:
4536:
4462:
4222:
4002:
3949:
2891:
1328:
1247:
803:
is the angle through which the pulley turns in radians. When multiplying
682:
362:
275:
160:
144:
4621:(SI) has long been a source of controversy and confusion." In 1960, the
4531:
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)}
6364:
Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"".
5425:
4115:
2899:
457:
274:(SI) and is the standard unit of angular measure used in many areas of
6329:
6106:
5253:
5218:
5183:
4678:, a higher-dimensional analog of the radian which measures solid angle
4595:
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.
6988:
6903:
6128:(in French), Consultative Committee for Units, May 1980, pp. 6–7
5910:
5868:
5825:
5538:
5413:
4675:
4626:
6498:
6464:
6439:
5886:
5843:
5800:
6868:
6760:
6672:
6401:
Torrens, A B (1 January 1986). "On Angles and Angular Quantities".
6177:
6020:
5935:
5477:
5351:
4430:
3130:
3126:
897:
equal to 1 inverse radian (1 rad) in a fashion similar to the
6557:
5726:(in Latin). Translated by Bruce, Ian. Definition 6, paragraph 316.
4653:
CCU Working Group on Angles and Dimensionless Quantities in the SI
3627:, then the series would contain messy factors involving powers of
7056:
7041:
6983:
6913:
6908:
6858:
6343:
Journal of Research of the National Bureau of Standards Section B
5197:
Oberhofer, E. S. (March 1992). "What happens to the 'radians'?".
4567:. In 1874, after a consultation with James Thomson, Muir adopted
4544:
3987:
3864:
2895:
1360:
227:
55:
6233:
6215:
5584:
Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi
4934:
4643:, but turned it down to avoid an upheaval to current practice.
4617:
As Paul Quincey et al. writes, "the status of angles within the
3113:
7071:
6883:
6863:
6853:
6795:
6274:
859:
289:, defined in the SI as 1 rad = 1 and expressed in terms of the
65:
61:
5050:, p. 151: "One radian corresponds to the angle for which
4980:, p. 151: "One radian corresponds to the angle for which
7086:
7081:
7066:
6943:
6938:
6933:
6918:
6873:
3995:
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,}
2629:
Conversely, to convert from degrees to radians, multiply by
7061:
7003:
6998:
6806:
6085:
Nelson, Robert A. (March 1984). "The supplementary units".
4006:
932:
4735:
3952:(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
7036:
6963:
6953:
5936:"Earliest Known Uses of Some of the Words of Mathematics"
4010:
2902:
or 400). To convert from radians to gradians multiply by
448: radians is equal to 360 degrees. The relation
6741:
5544:
5431:
5419:
5311:
5306:
4965:"ISO 80000-3:2006 Quantities and Units - Space and Time"
4808:
4547:. He had used the term as early as 1871, while in 1869,
3937:) or newton-metre per radian (for torsional stiffness).
3912:, whilst if the phase angle difference of two waves is (
3129:
and most other branches of mathematics beyond practical
2320:. Thus, to convert from radians to degrees, multiply by
5738:
Plane Trigonometry: For the Use of Colleges and Schools
5231:
4841:. Vol. Part I: Plane Trigonometry. New York, USA:
3957:
94:
74:
6539:
Mohr, Peter J; Phillips, William D (1 February 2015).
5610:. Springer Science & Business Media. p. 134.
5446:
5444:
5442:
5440:
5403:
5401:
5388:
5386:
4575:
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.}
2937:, and to convert from gradians to radians multiply by
773:. In SI 2019, the SI radian is defined accordingly as
603:
595:
550:
532:
494:
466:
5630:
4771:
4710:
3867:
when angular measurements are required. For example,
3640:
3492:
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:
405:
371:
322:
3930:
an integer, they are considered to be in antiphase.
6005:
5631:O'Connor, J. J.; Robertson, E. F. (February 2005).
5437:
5398:
5383:
4921:Protter, Murray H.; Morrey, Charles B. Jr. (1970),
4800:{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}}
3875:(rad/s). One revolution per second corresponds to 2
1231:{\displaystyle x=\eta \theta =\theta /{\text{rad}}}
5765:
5607:The Mathematics of Measurement: A Critical History
4799:
4722:
4523:becoming widespread, the unit was commonly called
3836:
3594:
3460:
3405:
3336:
3262:
3203:
3092:
3029:
2962:
2929:
2882:
2841:
2777:
2709:
2658:
2618:
2539:
2474:
2404:
2351:
2312:
1282:
1262:
1238:is the angle in radians. The capitalized function
1230:
1185:
633:
581:
518:
426:
384:
341:
6151:International Bureau of Weights and Measures 2019
6139:International Bureau of Weights and Measures 2019
5048:International Bureau of Weights and Measures 2019
5006:, the radian can be defined in terms of the area
4978:International Bureau of Weights and Measures 2019
4952:International Bureau of Weights and Measures 2019
4893:International Bureau of Weights and Measures 2019
4878:International Bureau of Weights and Measures 2019
4866:International Bureau of Weights and Measures 2019
7153:
5669:(in Latin). Cambridge, England. pp. 94–95.
5558:"UnityDimensions—Wolfram Language Documentation"
4700:Other proposals include the abbreviation "rad" (
3998:beams is also usually measured in milliradians.
3159:
1290:if it is clear that the complete form is meant.
5959:, p. 6, London: Longmans, Green, and Co., 1890
4005:is an approximation of the milliradian used by
3886:is often radian per second per second (rad/s).
799:is the radius of the pulley in centimeters and
5715:Theoria Motus Corporum Solidorum seu Rigidorum
5146:Minds-on Physics: Advanced topics in mechanics
667:International Organization for Standardization
6822:
5720:Theory of the motion of solid or rigid bodies
4920:
4136: rad (around 4.8481 microradians).
6744:"On the dimension of angles and their units"
6538:
6475:"Dimensional angles and universal constants"
6472:
6275:International Bureau of Weights and Measures
5772:. Vol. 2. Dover Publications. pp.
5462:
5336:
5317:
5286:
5003:
4743:
4717:
4711:
3908: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
30:"㎭" redirects here. Not to be confused with
7113:Historical definitions of the SI base units
5590:]. Berlin: Akademie Verlag. p. 40.
5515:
4830:
4828:
6829:
6815:
6473:Lévy-Leblond, Jean-Marc (September 1998).
6437:
5332:
5330:
5328:
5326:
5282:
5082:Angular amplitude of swing No dimensions.
5080:. New Haven : Yale University Press.
4739:
3948:for submultiples are used with radians. A
54:
6777:
6759:
6671:
6574:
6556:
6463:
6354:
6307:
6176:
6080:
6078:
6019:
5867:
5824:
5588:Treatise on the Circumference of al-Kashi
5476:
5350:
5266:
5196:
5119:
4861:
4859:
4834:
4701:
4612:
4497:
2659:{\displaystyle {\pi }/{180}{\text{ rad}}}
1323:units library defines angle units with a
6339:"Angle as a fourth fundamental quantity"
5744:, Cambridge and London: MacMillan, 1864
5599:
5597:
5582:Luckey, Paul (1953) . Siggel, A. (ed.).
5516:Schabel, Matthias C.; Watanabe, Steven.
5073:
4825:
3940:
3112:
858:(N⋅m/rad), and not in the quantities of
839:American Association of Physics Teachers
6704:
6657:
6583:
6400:
6162:
5841:
5663:"Editoris notæ ad Harmoniam mensurarum"
5450:
5407:
5392:
5323:
5302:
5298:
5278:
5142:
5038:), in which case it has the units m⋅m."
5002:, p. 844: "Also, as alluded to in
4999:
4923:College Calculus with Analytic Geometry
4762:
4723:{\displaystyle \langle \theta \rangle }
4461:radian measure is normally credited to
3475:is the angle expressed in radians, the
691:
14:
7154:
6505:
6336:
6280:The International System of Units (SI)
6084:
6075:
5933:
5760:
5684:
5581:
5290:
5270:
5169:
5093:
4856:
4751:
4731:
4631:Consultative Committee for Units (CCU)
4437:as units, where one diameter part was
342:{\displaystyle \theta ={\frac {s}{r}}}
6810:
6620:
5660:
5603:
5594:
5294:
4150:
4147:
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
6506:Foster, Marcus P (1 December 2010).
6363:
5884:
5798:
5274:
4009:and other military organizations in
1263:{\displaystyle \operatorname {Sin} }
589:. This can be further simplified to
5887:"The Term "Radian" in Trigonometry"
5844:"The Term "Radian" in Trigonometry"
5801:"The Term "Radian" in Trigonometry"
5637:The MacTutor History of Mathematics
5604:Roche, John J. (21 December 1998).
4424:
3871:is typically expressed in the unit
2930:{\displaystyle 200^{\text{g}}/\pi }
2862:
427:{\displaystyle {\frac {2\pi r}{r}}}
353:is the subtended angle in radians,
308:One radian is defined as the angle
24:
6440:"Angles—Let's treat them squarely"
4792:
4774:
2281:As stated, one radian is equal to
2276:
1304:is assumed to hold, or similarly,
25:
7188:
6788:
6061:Le Système international d'unités
5768:History of Mathematical Notations
5711:
5688:Roger Cotes - Natural Philosopher
5074:Bridgman, Percy Williams (1922).
3413:, the evaluation of the integral
829:, radians appear in the units of
697:
385:{\displaystyle {\frac {\pi }{2}}}
7136:
7135:
6794:
6337:Romain, Jacques E. (July 1962).
6235:Consultative Committee for Units
6217:Consultative Committee for Units
5988:
4606:
6541:"Dimensionless units in the SI"
6438:Brownstein, K. R. (July 1997).
6303:from the original on 8 May 2021
6245:
6227:
6209:
6156:
6113:
6067:(in French), 1970, p. 12,
6052:
5999:
5982:
5970:
5949:
5927:
5790:
5754:
5730:
5705:
5685:Gowing, Ronald (27 June 2002).
5678:
5654:
5624:
5575:
5550:
5509:
5456:
5260:
5225:
5190:
5163:
5136:
5094:Prando, Giacomo (August 2020).
5087:
5067:
5041:
4993:
4694:
4555:, vacillated between the terms
3613:but expressed in degrees, i.e.
1246:is the traditional function on
722:is subtended angle in radians,
7123:System of units of measurement
5691:. Cambridge University Press.
4971:
4957:
4914:
4898:
4871:
4603:which is analogously defined.
4068:; for example Sweden used the
4017:. 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:
5957:Longmans' School Trigonometry
5934:Miller, Jeff (Nov 23, 2009).
5149:. Kendall Hunt. p. 262.
4818:
4619:International System of Units
4577:Longmans' School Trigonometry
4143:SI multiples of radian (rad)
3882:Similarly, the unit used for
3863:The radian is widely used in
2894:, which is by definition 400
899:introduction of the constant
750:the area of a circular sector
303:
272:International System of Units
6836:
5143:Leonard, William J. (1999).
4593: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:
6479:American Journal of Physics
6444:American Journal of Physics
6310:American Journal of Physics
5533:Angles are treated as units
4659:
4613:§ Dimensional analysis
4498:§ Dimensional analysis
3984:(stadiametric) rangefinding
68:of that circle subtends an
10:
7193:
6643:10.1088/0026-1394/53/3/991
6606:10.1088/0026-1394/53/2/840
6524:10.1088/0026-1394/47/6/R01
6423:10.1088/0026-1394/22/1/002
6386:10.1088/0026-1394/18/1/002
5665:. In Smith, Robert (ed.).
5633:"Biography of Roger Cotes"
5518:"Boost.Units FAQ – 1.79.0"
5369:10.1088/0026-1394/53/3/998
4610:
4419:
3858:
2890:radians, which equals one
1316:in mathematical formulas.
1297:system where the equation
695:
36:
29:
7131:
7095:
7017:
6892:
6844:
6576:10.1088/0026-1394/52/1/40
6241:(Report). pp. 16–17.
6223:(Report). pp. 18–20.
6122:Report of the 7th meeting
5121:10.1038/s41567-020-0997-3
4925:(2nd ed.), Reading:
4433:(c. 1400) used so-called
4170:
4167:
4164:
4161:
4158:
4155:
4110:) and nanoradians (nrad,
223:
190:
156:
140:
136:
118:
110:
100:
88:
83:
53:
48:
7077:minute and second of arc
6779:10.1088/1681-7575/ac7bc2
6727:10.1088/1681-7575/abe0fc
6690:10.1088/1681-7575/ac023f
6237:(21–23 September 2021).
6195:10.1088/1681-7575/ab3fbf
6038:10.1088/1681-7575/ab27d7
5978:link at Internet Archive
5495:10.1088/1681-7575/aa7160
5318:Mohr & Phillips 2015
5004:Mohr & Phillips 2015
4687:
4671:Minute and second of arc
4597:Papers on Space Analysis
4583:when published in 1890.
4579:still called the radian
4553: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
7118:2019 revision of the SI
5994:Geometry/Unified Angles
5842:Thomson, James (1910).
3142:trigonometric functions
1312:allows the omission of
39:Radian (disambiguation)
4843:Henry Holt and Company
4801:
4724:
4527:of an angle. The term
3838:
3596:
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:
428:
386:
343:
6356:10.6028/jres.066B.012
5661:Cotes, Roger (1722).
5562:reference.wolfram.com
4802:
4734:), and the constants
4725:
3941:Prefixes and variants
3839:
3597:
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:
429:
387:
344:
280:SI supplementary unit
7177:Units of plane angle
7019:Other accepted units
6803:at Wikimedia Commons
5885:Muir, Thos. (1910).
5799:Muir, Thos. (1910).
5077:Dimensional analysis
4769:
4708:
4588:Alexander Macfarlane
3982:manufacturers using
3891:dimensional analysis
3884:angular acceleration
3879:radians per second.
3849:exponential function
3638:
3490:
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:
403:
369:
320:
37:For other uses, see
32:Rad (radiation unit)
7103:Conversion of units
6770:2022Metro..59e3001M
6719:2021Metro..58e2001L
6682:2021Metro..58e3002Q
6635:2016Metro..53..991M
6598:2016Metro..53..840Q
6567:2015Metro..52...40M
6491:1998AmJPh..66..814L
6456:1997AmJPh..65..605B
6415:1986Metro..22....1T
6378:1982Metro..18....1E
6322:1936AmJPh...4..175B
6219:(11–12 June 2013).
6187:2019Metro..56f5009K
6141:, pp. 174–175.
6099:1984PhTea..22..188N
6087:The Physics Teacher
6030:2019Metro..56d3001Q
5903:1910Natur..83..459M
5860:1910Natur..83..217T
5817:1910Natur..83..156M
5667:Harmonia mensurarum
5487:2017Metro..54..454Q
5361:2016Metro..53..998Q
5246:1993PhTea..31...84A
5234:The Physics Teacher
5211:1992PhTea..30..170O
5199:The Physics Teacher
5172:The Physics Teacher
5112:2020NatPh..16..888P
4467:Harmonia mensurarum
4144:
4033:of a circle and is
3889:For the purpose of
3851:(see, for example,
1341:
856:torsional stiffness
795:centimeters, where
726:is arc length, and
698:§ As a SI unit
397:complete revolution
357:is arc length, and
282:and is currently a
252: grad ≈ 63.662
131:... is equal to ...
84:General information
6896:with special names
5955:Frederick Sparks,
4967:. 17 January 2017.
4797:
4720:
4519:Prior to the term
4495:. As discussed in
4142:
4087:and the USSR used
3935:angular wavelength
3834:
3592:
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:
424:
382:
339:
233:
225:
200:
192:
166:
158:
150:
142:
7149:
7148:
7027:astronomical unit
6799:Media related to
6330:10.1119/1.1999110
6293:978-92-822-2272-0
6107:10.1119/1.2341516
5897:(2120): 459–460.
5736:Isaac Todhunter,
5712:Euler, Leonhard.
5698:978-0-521-52649-4
5617:978-0-387-91581-4
5287:Lévy-Leblond 1998
5254:10.1119/1.2343667
5219:10.1119/1.2343500
5184:10.1119/1.2343535
5156:978-0-7872-5412-4
5096:"A spectral unit"
4929:, p. APP-4,
4795:
4744:Lévy-Leblond 1998
4666:Angular frequency
4471:semicircumference
4417:
4416:
3904:) radians, where
3873:radian per second
3823:
3802:
3788:
3770:
3749:
3735:
3717:
3696:
3682:
3661:
3581:
3556:
3531:
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:
422:
380:
337:
266:, is the unit of
256:
255:
119:Conversions
16:(Redirected from
7184:
7167:SI derived units
7139:
7138:
6831:
6824:
6817:
6808:
6807:
6798:
6783:
6781:
6763:
6738:
6701:
6675:
6654:
6617:
6580:
6578:
6560:
6535:
6502:
6469:
6467:
6434:
6397:
6360:
6358:
6333:
6304:
6302:
6286:(9th ed.),
6285:
6265:
6264:
6262:
6260:
6249:
6243:
6242:
6231:
6225:
6224:
6213:
6207:
6206:
6180:
6160:
6154:
6148:
6142:
6136:
6130:
6129:
6127:
6117:
6111:
6110:
6082:
6073:
6072:
6066:
6056:
6050:
6049:
6023:
6003:
5997:
5992:
5986:
5980:
5974:
5968:
5953:
5947:
5946:
5944:
5942:
5931:
5925:
5922:
5911:10.1038/083459d0
5881:
5871:
5869:10.1038/083217c0
5838:
5828:
5826:10.1038/083156a0
5794:
5788:
5787:
5771:
5758:
5752:
5734:
5728:
5727:
5725:
5709:
5703:
5702:
5682:
5676:
5675:
5658:
5652:
5651:
5649:
5648:
5639:. Archived from
5628:
5622:
5621:
5601:
5592:
5591:
5579:
5573:
5572:
5570:
5568:
5554:
5548:
5545:Mohr et al. 2022
5542:
5536:
5535:
5530:
5528:
5513:
5507:
5506:
5480:
5460:
5454:
5448:
5435:
5432:Mohr et al. 2022
5429:
5423:
5420:Mohr et al. 2022
5417:
5411:
5405:
5396:
5390:
5381:
5380:
5354:
5334:
5321:
5315:
5309:
5307:Mohr et al. 2022
5264:
5258:
5257:
5229:
5223:
5222:
5194:
5188:
5187:
5167:
5161:
5160:
5140:
5134:
5133:
5123:
5091:
5085:
5084:
5071:
5065:
5063:
5059:
5045:
5039:
5037:
5030:
5028:
5027:
5024:
5021:
4997:
4991:
4989:
4975:
4969:
4968:
4961:
4955:
4949:
4938:
4937:
4918:
4912:
4911:
4902:
4896:
4890:
4881:
4875:
4869:
4863:
4854:
4853:
4851:
4850:
4832:
4812:
4809:Mohr et al. 2022
4806:
4804:
4803:
4798:
4796:
4791:
4783:
4778:
4777:
4729:
4727:
4726:
4721:
4704:), the notation
4698:
4642:
4601:hyperbolic angle
4581:circular measure
4525:circular measure
4515:
4494:
4452:
4450:
4449:
4446:
4443:
4425:Pre-20th century
4145:
4141:
4135:
4133:
4132:
4129:
4126:
4125:
4113:
4109:
4102:
4100:
4099:
4096:
4093:
4083:
4081:
4080:
4077:
4074:
4067:
4065:
4064:
4063:
4058:
4055:
4048:
4046:
4045:
4042:
4039:
4032:
4030:
4029:
4026:
4023:
3980:telescopic sight
3977:
3975:
3974:
3971:
3968:
3960:
3955:
3926:) radians, with
3925:
3924:
3920:
3903:
3878:
3869:angular velocity
3843:
3841:
3840:
3835:
3824:
3822:
3814:
3813:
3804:
3800:
3799:
3798:
3793:
3789:
3781:
3771:
3769:
3761:
3760:
3751:
3747:
3746:
3745:
3740:
3736:
3728:
3718:
3716:
3708:
3707:
3698:
3694:
3693:
3692:
3687:
3683:
3675:
3662:
3654:
3630:
3626:
3621:
3601:
3599:
3598:
3593:
3582:
3580:
3572:
3571:
3562:
3557:
3555:
3547:
3546:
3537:
3532:
3530:
3522:
3521:
3512:
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:
2686:angle in degrees
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:
2379:angle in radians
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:
447:
440:
433:
431:
430:
425:
423:
418:
407:
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:
7192:
7191:
7187:
7186:
7185:
7183:
7182:
7181:
7152:
7151:
7150:
7145:
7127:
7108:Metric prefixes
7091:
7013:
6895:
6888:
6840:
6835:
6791:
6786:
6499:10.1119/1.18964
6465:10.1119/1.18616
6300:
6294:
6283:
6277:(20 May 2019),
6269:
6268:
6258:
6256:
6251:
6250:
6246:
6232:
6228:
6214:
6210:
6161:
6157:
6149:
6145:
6137:
6133:
6125:
6119:
6118:
6114:
6083:
6076:
6064:
6058:
6057:
6053:
6004:
6000:
5987:
5983:
5975:
5971:
5954:
5950:
5940:
5938:
5932:
5928:
5795:
5791:
5784:
5762:Cajori, Florian
5759:
5755:
5735:
5731:
5723:
5710:
5706:
5699:
5683:
5679:
5659:
5655:
5646:
5644:
5629:
5625:
5618:
5602:
5595:
5580:
5576:
5566:
5564:
5556:
5555:
5551:
5543:
5539:
5526:
5524:
5514:
5510:
5461:
5457:
5449:
5438:
5434:, pp. 8–9.
5430:
5426:
5418:
5414:
5406:
5399:
5391:
5384:
5345:(3): 998–1002.
5335:
5324:
5316:
5312:
5283:Brownstein 1997
5265:
5261:
5230:
5226:
5195:
5191:
5168:
5164:
5157:
5141:
5137:
5092:
5088:
5072:
5068:
5061:
5051:
5046:
5042:
5025:
5022:
5019:
5018:
5016:
5011:
4998:
4994:
4981:
4976:
4972:
4963:
4962:
4958:
4950:
4941:
4919:
4915:
4904:
4903:
4899:
4891:
4884:
4876:
4872:
4864:
4857:
4848:
4846:
4833:
4826:
4821:
4816:
4815:
4784:
4782:
4773:
4772:
4770:
4767:
4766:
4760:
4740:Brownstein 1997
4709:
4706:
4705:
4699:
4695:
4690:
4662:
4640:
4634:
4615:
4609:
4541:Queen's College
4502:
4482:
4457:The concept of
4447:
4444:
4441:
4440:
4438:
4427:
4422:
4139:
4130:
4127:
4123:
4122:
4121:
4119:
4111:
4107:
4097:
4094:
4091:
4090:
4088:
4078:
4075:
4072:
4071:
4069:
4061:
4059:
4056:
4053:
4052:
4050:
4043:
4040:
4037:
4036:
4034:
4027:
4024:
4021:
4020:
4018:
3972:
3969:
3966:
3965:
3963:
3958:
3954:1 rad = 10 mrad
3953:
3946:Metric prefixes
3943:
3922:
3918:
3913:
3901:
3876:
3861:
3853:Euler's formula
3815:
3809:
3805:
3803:
3794:
3780:
3776:
3775:
3762:
3756:
3752:
3750:
3741:
3727:
3723:
3722:
3709:
3703:
3699:
3697:
3688:
3674:
3670:
3669:
3653:
3639:
3636:
3635:
3628:
3619:
3614:
3609:were the angle
3573:
3567:
3563:
3561:
3548:
3542:
3538:
3536:
3523:
3517:
3513:
3511:
3491:
3488:
3487:
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:
442:
435:
408:
406:
404:
401:
400:
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:
7190:
7180:
7179:
7174:
7169:
7164:
7147:
7146:
7144:
7143:
7132:
7129:
7128:
7126:
7125:
7120:
7115:
7110:
7105:
7099:
7097:
7093:
7092:
7090:
7089:
7084:
7079:
7074:
7069:
7064:
7059:
7054:
7049:
7044:
7039:
7034:
7029:
7023:
7021:
7015:
7014:
7012:
7011:
7006:
7001:
6996:
6991:
6986:
6981:
6976:
6971:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6914:degree Celsius
6911:
6906:
6900:
6898:
6894:Derived units
6890:
6889:
6887:
6886:
6881:
6876:
6871:
6866:
6861:
6856:
6850:
6848:
6842:
6841:
6834:
6833:
6826:
6819:
6811:
6805:
6804:
6790:
6789:External links
6787:
6785:
6784:
6739:
6702:
6655:
6629:(3): 991–997.
6618:
6592:(2): 840–845.
6581:
6536:
6518:(6): R41–R51.
6503:
6485:(9): 814–815.
6470:
6450:(7): 605–614.
6435:
6398:
6361:
6334:
6316:(4): 175–179.
6305:
6292:
6270:
6267:
6266:
6244:
6226:
6208:
6155:
6153:, p. 179.
6143:
6131:
6112:
6093:(3): 188–193.
6074:
6051:
5998:
5981:
5969:
5967:(1891 edition)
5948:
5926:
5924:
5923:
5882:
5839:
5789:
5782:
5753:
5729:
5704:
5697:
5677:
5653:
5623:
5616:
5593:
5574:
5549:
5537:
5508:
5471:(4): 454–460.
5455:
5436:
5424:
5412:
5397:
5382:
5322:
5310:
5267:Brinsmade 1936
5259:
5224:
5205:(3): 170–171.
5189:
5178:(5): 260–261.
5162:
5155:
5135:
5100:Nature Physics
5086:
5066:
5040:
4992:
4970:
4956:
4954:, p. 151.
4939:
4927:Addison-Wesley
4913:
4897:
4895:, p. 137.
4882:
4870:
4855:
4823:
4822:
4820:
4817:
4814:
4813:
4794:
4790:
4787:
4781:
4776:
4758:
4719:
4716:
4713:
4702:Brinsmade 1936
4692:
4691:
4689:
4686:
4685:
4684:
4679:
4673:
4668:
4661:
4658:
4638:
4608:
4605:
4551:, then of the
4478:Leonhard Euler
4435:diameter parts
4426:
4423:
4421:
4418:
4415:
4414:
4411:
4408:
4405:
4402:
4399:
4395:
4394:
4391:
4388:
4385:
4382:
4379:
4375:
4374:
4371:
4368:
4365:
4362:
4359:
4355:
4354:
4351:
4348:
4345:
4342:
4339:
4335:
4334:
4331:
4328:
4325:
4322:
4319:
4315:
4314:
4311:
4308:
4305:
4302:
4299:
4295:
4294:
4291:
4288:
4285:
4282:
4279:
4275:
4274:
4271:
4268:
4265:
4262:
4259:
4255:
4254:
4251:
4248:
4245:
4242:
4239:
4235:
4234:
4231:
4228:
4225:
4220:
4217:
4213:
4212:
4209:
4206:
4203:
4200:
4197:
4193:
4192:
4189:
4186:
4183:
4180:
4177:
4173:
4172:
4169:
4166:
4163:
4160:
4157:
4153:
4152:
4149:
3942:
3939:
3860:
3857:
3845:
3844:
3833:
3830:
3827:
3821:
3818:
3812:
3808:
3797:
3792:
3787:
3784:
3779:
3774:
3768:
3765:
3759:
3755:
3744:
3739:
3734:
3731:
3726:
3721:
3715:
3712:
3706:
3702:
3691:
3686:
3681:
3678:
3673:
3668:
3665:
3660:
3657:
3652:
3649:
3646:
3643:
3603:
3602:
3591:
3588:
3585:
3579:
3576:
3570:
3566:
3560:
3554:
3551:
3545:
3541:
3535:
3529:
3526:
3520:
3516:
3510:
3507:
3504:
3501:
3498:
3495:
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:
421:
417:
414:
411:
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:
7189:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7162:Natural units
7160:
7159:
7157:
7142:
7134:
7133:
7130:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7100:
7098:
7094:
7088:
7085:
7083:
7080:
7078:
7075:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7047:degree of arc
7045:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7024:
7022:
7020:
7016:
7010:
7007:
7005:
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6970:
6967:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6901:
6899:
6897:
6891:
6885:
6882:
6880:
6877:
6875:
6872:
6870:
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6851:
6849:
6847:
6843:
6839:
6832:
6827:
6825:
6820:
6818:
6813:
6812:
6809:
6802:
6797:
6793:
6792:
6780:
6775:
6771:
6767:
6762:
6757:
6754:(5): 053001.
6753:
6749:
6745:
6740:
6736:
6732:
6728:
6724:
6720:
6716:
6713:(5): 052001.
6712:
6708:
6703:
6699:
6695:
6691:
6687:
6683:
6679:
6674:
6669:
6666:(5): 053002.
6665:
6661:
6656:
6652:
6648:
6644:
6640:
6636:
6632:
6628:
6624:
6619:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6587:
6582:
6577:
6572:
6568:
6564:
6559:
6554:
6550:
6546:
6542:
6537:
6533:
6529:
6525:
6521:
6517:
6513:
6509:
6504:
6500:
6496:
6492:
6488:
6484:
6480:
6476:
6471:
6466:
6461:
6457:
6453:
6449:
6445:
6441:
6436:
6432:
6428:
6424:
6420:
6416:
6412:
6408:
6404:
6399:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6367:
6362:
6357:
6352:
6348:
6344:
6340:
6335:
6331:
6327:
6323:
6319:
6315:
6311:
6306:
6299:
6295:
6289:
6282:
6281:
6276:
6272:
6271:
6254:
6248:
6240:
6236:
6230:
6222:
6218:
6212:
6204:
6200:
6196:
6192:
6188:
6184:
6179:
6174:
6171:(6): 065009.
6170:
6166:
6159:
6152:
6147:
6140:
6135:
6124:
6123:
6116:
6108:
6104:
6100:
6096:
6092:
6088:
6081:
6079:
6070:
6063:
6062:
6055:
6047:
6043:
6039:
6035:
6031:
6027:
6022:
6017:
6014:(4): 043001.
6013:
6009:
6002:
5995:
5991:
5985:
5979:
5973:
5966:
5962:
5958:
5952:
5937:
5930:
5920:
5916:
5912:
5908:
5904:
5900:
5896:
5892:
5888:
5883:
5879:
5875:
5870:
5865:
5861:
5857:
5854:(2112): 217.
5853:
5849:
5845:
5840:
5836:
5832:
5827:
5822:
5818:
5814:
5811:(2110): 156.
5810:
5806:
5802:
5797:
5796:
5793:
5785:
5783:0-486-67766-4
5779:
5775:
5770:
5769:
5763:
5757:
5751:
5747:
5743:
5739:
5733:
5721:
5717:
5716:
5708:
5700:
5694:
5690:
5689:
5681:
5673:
5668:
5664:
5657:
5643:on 2012-10-19
5642:
5638:
5634:
5627:
5619:
5613:
5609:
5608:
5600:
5598:
5589:
5585:
5578:
5563:
5559:
5553:
5546:
5541:
5534:
5523:
5522:www.boost.org
5519:
5512:
5504:
5500:
5496:
5492:
5488:
5484:
5479:
5474:
5470:
5466:
5459:
5452:
5447:
5445:
5443:
5441:
5433:
5428:
5421:
5416:
5409:
5404:
5402:
5394:
5389:
5387:
5378:
5374:
5370:
5366:
5362:
5358:
5353:
5348:
5344:
5340:
5333:
5331:
5329:
5327:
5319:
5314:
5308:
5304:
5300:
5296:
5292:
5288:
5284:
5280:
5276:
5272:
5268:
5263:
5255:
5251:
5247:
5243:
5239:
5235:
5228:
5220:
5216:
5212:
5208:
5204:
5200:
5193:
5185:
5181:
5177:
5173:
5166:
5158:
5152:
5148:
5147:
5139:
5131:
5127:
5122:
5117:
5113:
5109:
5105:
5101:
5097:
5090:
5083:
5079:
5078:
5070:
5058:
5054:
5049:
5044:
5036:
5033:
5014:
5010:of a sector (
5009:
5005:
5001:
4996:
4988:
4984:
4979:
4974:
4966:
4960:
4953:
4948:
4946:
4944:
4936:
4932:
4928:
4924:
4917:
4909:
4908:
4901:
4894:
4889:
4887:
4879:
4874:
4867:
4862:
4860:
4844:
4840:
4839:
4831:
4829:
4824:
4810:
4788:
4785:
4779:
4764:
4757:
4753:
4749:
4745:
4741:
4737:
4733:
4714:
4703:
4697:
4693:
4683:
4680:
4677:
4674:
4672:
4669:
4667:
4664:
4663:
4657:
4654:
4648:
4644:
4637:
4632:
4628:
4624:
4620:
4614:
4607:As an SI unit
4604:
4602:
4598:
4594:
4589:
4584:
4582:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4534:
4533:James Thomson
4530:
4526:
4522:
4517:
4513:
4509:
4505:
4500:
4499:
4493:
4489:
4485:
4479:
4474:
4472:
4468:
4464:
4460:
4455:
4436:
4432:
4413:quettaradian
4412:
4409:
4406:
4404:quectoradian
4403:
4400:
4397:
4396:
4392:
4389:
4386:
4383:
4380:
4377:
4376:
4372:
4369:
4366:
4363:
4360:
4357:
4356:
4352:
4349:
4346:
4343:
4340:
4337:
4336:
4332:
4329:
4326:
4323:
4320:
4317:
4316:
4312:
4309:
4306:
4303:
4300:
4297:
4296:
4292:
4289:
4286:
4283:
4280:
4277:
4276:
4272:
4269:
4266:
4263:
4260:
4257:
4256:
4252:
4249:
4246:
4243:
4240:
4237:
4236:
4232:
4229:
4226:
4224:
4221:
4218:
4215:
4214:
4210:
4207:
4204:
4201:
4198:
4195:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4174:
4154:
4148:Submultiples
4146:
4140:
4137:
4117:
4104:
4086:
4016:
4012:
4008:
4004:
3999:
3997:
3993:
3989:
3985:
3981:
3961:
3956:. There are 2
3951:
3947:
3938:
3936:
3931:
3929:
3916:
3911:
3907:
3899:
3894:
3892:
3887:
3885:
3880:
3874:
3870:
3866:
3856:
3854:
3850:
3831:
3828:
3825:
3819:
3816:
3810:
3806:
3795:
3790:
3785:
3782:
3777:
3772:
3766:
3763:
3757:
3753:
3742:
3737:
3732:
3729:
3724:
3719:
3713:
3710:
3704:
3700:
3689:
3684:
3679:
3676:
3671:
3666:
3663:
3658:
3655:
3650:
3647:
3644:
3641:
3634:
3633:
3632:
3624:
3617:
3612:
3608:
3589:
3586:
3583:
3577:
3574:
3568:
3564:
3558:
3552:
3549:
3543:
3539:
3533:
3527:
3524:
3518:
3514:
3508:
3505:
3502:
3499:
3496:
3493:
3486:
3485:
3484:
3482:
3479:for sin
3478:
3477:Taylor series
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:
446:
439:
419:
415:
412:
409:
398:
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:
7052:electronvolt
6973:
6751:
6747:
6710:
6706:
6663:
6659:
6626:
6622:
6589:
6585:
6551:(1): 40–47.
6548:
6544:
6515:
6511:
6482:
6478:
6447:
6443:
6406:
6402:
6369:
6365:
6346:
6342:
6313:
6309:
6279:
6257:. Retrieved
6247:
6229:
6211:
6168:
6164:
6158:
6146:
6134:
6121:
6115:
6090:
6086:
6068:
6060:
6054:
6011:
6007:
6001:
5996:at Wikibooks
5984:
5972:
5956:
5951:
5939:. Retrieved
5929:
5894:
5890:
5851:
5847:
5808:
5804:
5792:
5767:
5756:
5737:
5732:
5719:
5714:
5707:
5687:
5680:
5670:
5666:
5656:
5645:. Retrieved
5641:the original
5636:
5626:
5606:
5587:
5583:
5577:
5565:. Retrieved
5561:
5552:
5547:, p. 3.
5540:
5532:
5525:. Retrieved
5521:
5511:
5468:
5464:
5458:
5451:Quincey 2021
5427:
5422:, p. 6.
5415:
5408:Torrens 1986
5393:Quincey 2016
5342:
5338:
5313:
5303:Leonard 2021
5299:Quincey 2021
5279:Torrens 1986
5262:
5240:(2): 84–87.
5237:
5233:
5227:
5202:
5198:
5192:
5175:
5171:
5165:
5145:
5138:
5103:
5099:
5089:
5081:
5076:
5069:
5056:
5052:
5043:
5034:
5031:
5012:
5007:
5000:Quincey 2016
4995:
4986:
4982:
4973:
4959:
4922:
4916:
4906:
4900:
4873:
4847:. Retrieved
4838:Trigonometry
4837:
4763:Quincey 2021
4755:
4747:
4696:
4682:Trigonometry
4652:
4649:
4645:
4635:
4616:
4596:
4585:
4580:
4576:
4572:
4568:
4564:
4560:
4556:
4535:(brother of
4528:
4524:
4520:
4518:
4511:
4507:
4503:
4496:
4491:
4487:
4483:
4475:
4466:
4458:
4456:
4434:
4428:
4407:10 rad
4398:10 rad
4393:ronnaradian
4387:10 rad
4384:rontoradian
4378:10 rad
4373:yottaradian
4367:10 rad
4364:yoctoradian
4358:10 rad
4353:zettaradian
4347:10 rad
4344:zeptoradian
4338:10 rad
4327:10 rad
4318:10 rad
4307:10 rad
4304:femtoradian
4298:10 rad
4287:10 rad
4278:10 rad
4267:10 rad
4258:10 rad
4247:10 rad
4244:microradian
4238:10 rad
4227:10 rad
4216:10 rad
4211:hectoradian
4205:10 rad
4202:centiradian
4196:10 rad
4185:10 rad
4176:10 rad
4138:
4105:
4084:
4000:
3944:
3932:
3927:
3914:
3905:
3897:
3895:
3888:
3881:
3862:
3846:
3622:
3615:
3610:
3606:
3604:
3480:
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:
444:
437:
394:
307:
291:SI base unit
263:
259:
257:
145:milliradians
130:
125:
101:Unit of
69:
60:An arc of a
43:
6372:(1): 1–12.
5291:Foster 2010
5271:Romain 1962
4752:Foster 2010
4732:Romain 1962
4571:. The name
4549:Thomas Muir
4537:Lord Kelvin
4463:Roger Cotes
4324:attoradian
4313:petaradian
4293:teraradian
4284:picoradian
4273:gigaradian
4264:nanoradian
4253:megaradian
4233:kiloradian
4223:milliradian
4191:decaradian
4182:deciradian
4118:, which is
4112:10 rad
4108:10 rad
4003:angular mil
3950: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
7156:Categories
6846:Base units
6761:2203.12392
6748:Metrologia
6707:Metrologia
6673:2108.05704
6660:Metrologia
6623:Metrologia
6586:Metrologia
6545:Metrologia
6512:Metrologia
6409:(1): 1–7.
6403:Metrologia
6366:Metrologia
6178:1810.12057
6165:Metrologia
6021:1909.08389
6008:Metrologia
5647:2006-04-21
5478:1705.03765
5465:Metrologia
5352:1604.02373
5339:Metrologia
5295:Mills 2016
5106:(8): 888.
4849:2017-08-12
4819:References
4611:See also:
4333:exaradian
4168:SI symbol
4159:SI symbol
4151:Multiples
4116:arc second
3992: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
6989:steradian
6904:becquerel
6735:234036217
6698:236547235
6651:126032642
6614:125438811
6558:1409.2794
6532:117711734
6431:250801509
6394:250750831
6349:(3): 97.
6046:198428043
5965:877238863
5750:500022958
5503:119418270
5377:119294905
5275:Eder 1982
5130:225445454
5062:1 rad = 1
4793:Θ
4789:π
4718:⟩
4715:θ
4712:⟨
4676:Steradian
4627:steradian
4476:In 1765,
4015:targeting
3829:⋯
3783:π
3773:−
3730:π
3677:π
3667:−
3656:π
3645:
3587:⋯
3559:−
3509:−
3497:
3483: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
413:π
392:radians.
375:π
324:θ
314:magnitude
310:subtended
298:rad = m/m
152:1000 mrad
7141:Category
7096:See also
6869:kilogram
6838:SI units
6298:archived
6203:53627142
5764:(1929).
4935:76087042
4660:See also
4586:In 1893
4431:al-Kashi
3988: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
441:. Thus,
349:, where
228:gradians
78:radians.
7057:hectare
7042:decibel
6984:sievert
6979:siemens
6909:coulomb
6859:candela
6766:Bibcode
6715:Bibcode
6678:Bibcode
6631:Bibcode
6594:Bibcode
6563:Bibcode
6487:Bibcode
6452:Bibcode
6411:Bibcode
6374:Bibcode
6318:Bibcode
6259:26 June
6183:Bibcode
6095:Bibcode
6026:Bibcode
5941:Sep 30,
5919:3971449
5899:Bibcode
5878:3980250
5856:Bibcode
5835:3958702
5813:Bibcode
5774:147–148
5483:Bibcode
5357:Bibcode
5242:Bibcode
5207:Bibcode
5108:Bibcode
5060:, thus
5029:
5017:
4765:), and
4641:= 1 rad
4545:Belfast
4451:
4439:
4420:History
4134:
4131:648,000
4120:
4101:
4089:
4082:
4070:
4066:
4051:
4047:
4035:
4031:
4019:
4011:gunnery
3976:
3964:
3865:physics
3859: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
7072:minute
7032:dalton
6974:radian
6969:pascal
6959:newton
6884:second
6864:kelvin
6854:ampere
6801:Radian
6733:
6696:
6649:
6612:
6530:
6429:
6392:
6290:
6255:. BIPM
6201:
6044:
5963:
5917:
5891:Nature
5876:
5848:Nature
5833:
5805:Nature
5780:
5748:
5722:]
5695:
5614:
5567:1 July
5501:
5375:
5153:
5128:
4933:
4742:), ◁ (
4573:radian
4569:radian
4565:radian
4563:, and
4561:radial
4529:radian
4521:radian
4188:darad
4165:Value
4156:Value
4085:streck
3990:. The
3801:
3748:
3695:
3631:/180:
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
7087:tonne
7082:neper
7067:litre
7009:weber
6994:tesla
6949:lumen
6944:katal
6939:joule
6934:hertz
6929:henry
6919:farad
6874:metre
6756:arXiv
6731:S2CID
6694:S2CID
6668:arXiv
6647:S2CID
6610:S2CID
6553:arXiv
6528:S2CID
6427:S2CID
6390:S2CID
6301:(PDF)
6284:(PDF)
6199:S2CID
6173:arXiv
6126:(PDF)
6065:(PDF)
6042:S2CID
6016:arXiv
5915:S2CID
5874:S2CID
5831:S2CID
5742:p. 10
5724:(PDF)
5718:[
5586:[
5527:5 May
5499:S2CID
5473:arXiv
5373:S2CID
5347:arXiv
5126:S2CID
4688:Notes
4539:) at
4410:Qrad
4401:qrad
4390:Rrad
4381:rrad
4370:Yrad
4361:yrad
4350:Zrad
4341:zrad
4330:Erad
4321:arad
4310:Prad
4301:frad
4290:Trad
4281:prad
4270:Grad
4261:nrad
4250:Mrad
4241:μrad
4230:krad
4219:mrad
4208:hrad
4199:crad
4179:drad
4171:Name
4162:Name
3996:laser
3910:phase
3625:/ 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
434:, or
294:metre
268:angle
161:turns
105:angle
70:angle
7062:hour
7004:watt
6999:volt
6924:gray
6879:mole
6288:ISBN
6261:2022
5961:OCLC
5943:2011
5778:ISBN
5746:OCLC
5693:ISBN
5612:ISBN
5569:2022
5529:2022
5151:ISBN
4931:LCCN
4623:CGPM
4098:6000
4079:6300
4060:2000
4028:6400
4013:and
4007:NATO
4001:The
3973:6283
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°
395:One
258:The
186:turn
7037:day
6964:ohm
6954:lux
6774:doi
6723:doi
6686:doi
6639:doi
6602:doi
6571:doi
6520:doi
6495:doi
6460:doi
6419:doi
6382:doi
6351:doi
6347:66B
6326:doi
6191:doi
6103:doi
6034:doi
5907:doi
5864:doi
5821:doi
5491:doi
5365:doi
5250:doi
5215:doi
5180:doi
5116:doi
4754:),
4746:),
4557:rad
4459:the
3994:of
3986:in
3786:180
3733:180
3680:180
3659:180
3642:sin
3605:If
3494: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
7172:Pi
7158::
6772:.
6764:.
6752:59
6750:.
6746:.
6729:.
6721:.
6711:58
6709:.
6692:.
6684:.
6676:.
6664:58
6662:.
6645:.
6637:.
6627:53
6625:.
6608:.
6600:.
6590:53
6588:.
6569:.
6561:.
6549:52
6547:.
6543:.
6526:.
6516:47
6514:.
6510:.
6493:.
6483:66
6481:.
6477:.
6458:.
6448:65
6446:.
6442:.
6425:.
6417:.
6407:22
6405:.
6388:.
6380:.
6370:18
6368:.
6345:.
6341:.
6324:.
6312:.
6296:,
6197:.
6189:.
6181:.
6169:56
6167:.
6101:.
6091:22
6089:.
6077:^
6040:.
6032:.
6024:.
6012:56
6010:.
5913:.
5905:.
5895:83
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