What Times What Equals 360

SI derived unit of angle

Radian
An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.
General information
Unit organisation	SI
Unit of	Bending
Symbol	rad,c or r
Conversions
1 rad in ...	... is equal to ...
milliradians	thou mrad
turns	1 / 2π turn
degrees	180° / π ≈ 57.296°
gradians	200g / π ≈ 63.662chiliad

The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI), and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995).^[ane] The radian is defined in the SI as being a dimensionless unit with i rad = ane.^[2] Its symbol is accordingly ofttimes omitted, particularly in mathematical writing.

Definition

I radian is defined as the bending subtended from the eye of a circumvolve which intercepts an arc equal in length to the radius of the circle.^[3] More more often than not, the magnitude in radians of a subtended bending is equal to the ratio of the arc length to the radius of the circumvolve; that is, θ = southward/r , where θ is the subtended angle in radians, s is arc length, and r is radius. A right angle is exactly π / 2 radians.^[4]

The magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided past the radius, or iiπ r / r , or 2π. Thus iiπ radians is equal to 360 degrees, pregnant that i radian is equal to 180/π degrees ≈ 57.2957795130 82320 876... degrees.^[five]

The relation 2π rad = 360° tin be derived using the formula for arc length, ${\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}$ . Since radian is the mensurate of an angle that subtends an arc of a length equal to the radius of the circle, ${\textstyle ane=ii\pi \left({\tfrac {one{\text{ rad}}}{360^{\circ }}}\right)}$ . This can exist farther simplified to ${\textstyle ane={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}$ . Multiplying both sides past 360° gives 360° = twoπ rad.

Unit of measurement symbol

The International Bureau of Weights and Measures^[4] and International Organization for Standardization^[6] specify rad as the symbol for the radian. Alternative symbols that were in use in 1909 are ^c (the superscript letter of the alphabet c, for "circular measure out"), the alphabetic character r, or a superscript ^R,^[vii] but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence a value of ane.2 radians would exist written today as one.2 rad; archaic notations could include ane.two r, 1.2^rad, one.2^c, or 1.2^R.

In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are causeless, and when degrees are meant, the degree sign ° is used.

Dimensional analysis

The radian is defined equally θ = s/r , where θ is the subtended angle in radians, due south is arc length, and r is radius. One radian corresponds to the angle for which s=r , hence 1 radian = 1 one thousand / m .^[viii] Nevertheless, ${\displaystyle \mathrm {rad} }$ is simply to exist used to express angles, not to express ratios of lengths in general.^[4] A similar calculation using the area of a circular sector θ = 2A/r ² gives 1 radian as one chiliad² / m^two .^[ix] The key fact is that the radian is a dimensionless unit equal to 1. In SI 2019, the radian is defined accordingly as 1 rad = i.^[10] It is a long-established practice in mathematics and across all areas of science to brand employ of ${\displaystyle \mathrm {rad} =1}$ .^{[11] [12]} In 1993 the AAPT Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure out (rad), angular speed (rad/due south), athwart dispatch (rad/s^two), and torsional stiffness (Due north⋅k/rad), and not in the quantities of torque (North⋅grand) and angular momentum (kg⋅thou^two/s).^[13]

Giacomo Prando says "the current land of diplomacy leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations."^[14] For example, a mass hanging by a string from a pulley will ascent or driblet past y=rθ centimeters, where r is the radius of the pulley in centimeters and θ is the bending the caster turns in radians. When multiplying r past θ the unit of radians of disappears from the result. Similarly in the formula for the athwart velocity of a rolling wheel, ω=5/r , radians appear in the units of ω just not on the right hand side.^[15] Anthony French calls this miracle "a perennial trouble in the teaching of mechanics".^[16] Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units co-ordinate to convention and contextual cognition is "pedagogically unsatisfying".^[17]

At least a dozen scientists have made proposals to treat the radian equally a base unit of measure out defining its ain dimension of "angle", as early every bit 1936 and as recently as 2022.^{[18] [xix] [20]} Quincey's review of proposals outlines two classes of proposal. The beginning option changes the unit of measurement of a radius to meters per radian, but this is incompatible with dimensional assay for the expanse of a circumvolve, πr ⁱⁱ . The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, only requires "the modification of many familiar mathematical and concrete equations".^[21]

In detail Quincey identifies Torrens' proposal, to introduces a constant η equal to i inverse radian (1 rad⁻ⁱ) in a fashion similar to the introduction of the constant ε ₀.^{[21] [a]} With this change the formula for the angle subtended at the center of a circle, s = rθ , is modified to become s = ηrθ , and the Taylor series for the sine of an bending θ becomes:^{[20] [22]}

$${\displaystyle \operatorname {Sin} \theta =\sin _{\text{rad}}(\eta \theta )=\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{v!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots .}$$

The capitalized part ${\displaystyle \operatorname {Sin} }$ is the "complete" function that takes an statement with a dimension of bending and is independent of the units expressed,^[22] while ${\displaystyle \sin _{\text{rad}}}$ is the traditional function on pure numbers which assumes its argument is in radians.^[23] ${\displaystyle \operatorname {Sin} }$ can be denoted ${\displaystyle \sin }$ if information technology is clear that the complete grade is meant.^{[20] [24]}

SI can be considered relative to this framework every bit a natural unit arrangement where the equation η = ane is assumed to hold, or similarly one rad = i. This radian convention allows the omission of η in mathematical formulas.^[25]

A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional abiding is probable to forestall widespread use.^[twenty] Defining radian equally a base of operations unit of measurement may exist useful for software, where the disadvantage of longer equations is minimal.^[26] For example, the Boost units library defines bending units with a plane_angle dimension,^[27] and Mathematica'south unit organisation similarly considers angles to have an angle dimension.^{[28] [29]}

Conversions

Conversion of mutual angles

Turns	Radians	Degrees	Gradians
0 turn	0 rad	0°	01000
one / 24 plow	π / 12 rad	xv°	sixteen+ two / three m
i / sixteen plough	π / viii rad	22.five°	25grand
i / 12 turn	π / 6 rad	30°	33+ one / 3 grand
1 / x plow	π / 5 rad	36°	twoscoreg
1 / viii turn	π / iv rad	45°	50thou
one / 2π plough	ane rad	c. 57.3°	c. 63.7chiliad
ane / 6 plow	π / 3 rad	threescore°	66+ 2 / three m
one / 5 turn	2π / 5 rad	72°	80thou
i / 4 turn	π / 2 rad	90°	100g
1 / three plow	iiπ / 3 rad	120°	133+ 1 / 3 thousand
2 / five turn	4π / 5 rad	144°	160g
1 / 2 turn	π rad	180°	200g
3 / 4 plow	iiiπ / two rad	270°	300g
1 turn	2π rad	360°	400one thousand

Betwixt degrees

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

${\displaystyle {\text{bending in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}$

For example:

${\displaystyle ane{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$

${\displaystyle ii.5{\text{ rad}}=two.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}$

${\displaystyle {\frac {\pi }{iii}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}$

Conversely, to catechumen from degrees to radians, multiply by ${\displaystyle {\pi }/{180^{\circ }}}$ .

${\displaystyle {\text{angle in radians}}={\text{bending in degrees}}\cdot {\frac {\pi }{180^{\circ }}}}$

For example:

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

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

Radians can exist converted to turns (complete revolutions) past dividing the number of radians past 2π.

Between gradians

${\displaystyle 2\pi }$ radians equals one plough, which is by definition 400 gradians (400 gons or 400^yard). So, to convert from radians to gradians multiply by ${\displaystyle 200^{\text{g}}/\pi }$ , and to convert from gradians to radians multiply by ${\displaystyle \pi /200^{\text{yard}}}$ . For example,

${\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{1000}}}{\pi }}\approx 76.3944^{\text{g}}}$

${\displaystyle l^{\text{g}}=fifty^{\text{g}}\cdot {\frac {\pi }{200^{\text{g}}}}\approx 0.7854{\text{ rad}}}$

Usage

Mathematics

Some common angles, measured in radians. All the big polygons in this diagram are regular polygons.

In calculus and most other branches of mathematics across practical geometry, angles are universally measured in radians. This is considering radians have a mathematical "naturalness" that leads to a more than elegant formulation of a number of important results.

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

${\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}$

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

${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$ ^[5]

${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}$

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation ${\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}$ , the evaluation of the integral ${\displaystyle \textstyle \int {\frac {dx}{1+x^{ii}}},}$ then on). In all such cases, information technology is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

The trigonometric functions also take simple and elegant series expansions when radians are used. For case, when x is in radians, the Taylor series for sinten becomes:

${\displaystyle \sin x=x-{\frac {ten^{3}}{three!}}+{\frac {x^{5}}{5!}}-{\frac {x^{seven}}{seven!}}+\cdots .}$

If x were expressed in degrees, and then the serial would contain messy factors involving powers of π/180: if ten is the number of degrees, the number of radians is y = π x / 180, then

${\displaystyle \sin x_{\mathrm {deg} }=\sin y_{\mathrm {rad} }={\frac {\pi }{180}}ten-\left({\frac {\pi }{180}}\correct)^{iii}\ {\frac {x^{3}}{iii!}}+\left({\frac {\pi }{180}}\right)^{v}\ {\frac {x^{v}}{v!}}-\left({\frac {\pi }{180}}\correct)^{7}\ {\frac {10^{7}}{7!}}+\cdots .}$

In a similar spirit, mathematically important relationships between the sine and cosine functions and the exponential part (see, for example, Euler's formula) can be elegantly stated, when the functions' arguments are in radians (and messy otherwise).

Physics

The radian is widely used in physics when athwart measurements are required. For example, angular velocity is typically measured in radians per second (rad/southward). I revolution per 2d is equal to 2π radians per second.

Similarly, angular acceleration is often measured in radians per second per second (rad/southward²).

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

Too, the stage departure of ii waves can also be measured in radians. For case, if the phase difference of two waves is (n⋅twoπ) radians, where north is an integer, they are considered in stage, whilst if the phase difference of two waves is ( n⋅2π + π ), where n is an integer, they are considered in antiphase.

Prefixes and variants

Metric prefixes for submultiples are used with radians. A milliradian (mrad) is a thousandth of a radian (0.001 rad), i.e. 1 rad = xⁱⁱⁱ mrad. There are 2π × 1000 milliradians (≈ 6283.185 mrad) in a circumvolve. And then a milliradian is just nether 1 / 6283 of the angle subtended by a full circle. This unit of angular measurement of a circle is in mutual employ by scope sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

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

Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad, 10^{−half-dozen} rad) and nanoradians (nrad, 10^−nine rad) are used in astronomy, and tin can also exist used to measure the beam quality of lasers with ultra-low difference. More common is the arc 2nd, which is π / 648,000  rad (effectually 4.8481 microradians).

History

Pre-20th century

The idea of measuring angles by the length of the arc was in utilise by mathematicians quite early. For example, al-Kashi (c. 1400) used so-called diameter parts as units, where one diameter office was i / 60 radian. They also used sexagesimal subunits of the diameter role.^[thirty] Newton in 1672 spoke of "the angular quantity of a body's round motion", simply used information technology only as a relative measure out to develop an astronomical algorithm.^[31]

The concept of the radian measure is usually credited to Roger Cotes, who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, Harmonia mensurarum.^[32] In a affiliate of editorial comments, Smith gave what is probably the first published adding of one radian in degrees, citing a note of Cotes that has not survived. Smith described the radian in everything but proper name, and recognized its naturalness as a unit of angular measure.^{[33] [34]}

In 1765, Leonhard Euler implicitly adopted the radian as the angle unit for all equations involving rotation.^[31] Specifically, Euler defined angular velocity equally "The angular speed in rotational motion is the speed of that point, the distance of which from the axis of gyration is expressed past ane."^[35] Euler was probably the first to prefer this convention, referred to as the radian convention, which gives the simple formula for athwart velocity ω= v/r . Equally discussed in § Dimensional analysis, the radian convention has been widely adopted, and other conventions accept the drawback of requiring a dimensional constant, for case ω = v/(ηr) .^[25]

Prior to the term radian becoming widespread, the unit was unremarkably called circular measure of an angle.^[36] The term radian outset appeared in print on 5 June 1873, in test questions gear up by James Thomson (brother of Lord Kelvin) at Queen'southward College, Belfast. He had used the term every bit early as 1871, while in 1869, Thomas Muir, so of the University of St Andrews, vacillated between the terms rad, radial, and radian. In 1874, subsequently a consultation with James Thomson, Muir adopted radian.^{[37] [38] [39]} The name radian was not universally adopted for some time later this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.^[40]

As a SI unit

As Paul Quincey et al. writes, "the status of angles within the International System of Units (SI) has long been a source of controversy and defoliation."^[41] In 1960, the CGPM established the SI and the radian was classified equally a "supplementary unit" forth with the steradian. This special form was officially regarded "either equally base units or as derived units", as the CGPM could not attain a conclusion on whether the radian was a base of operations unit or a derived unit.^[42] Richard Nelson writes "This ambiguity [in the classification of the supplemental units] prompted a spirited word over their proper interpretation."^[43] In May 1980 the Consultative Commission for Units (CCU) considered a proposal for making radians an SI base unit, using a constant α ₀ = one rad,^{[44] [25]} but turned information technology downward to avoid an upheaval to current practise.^[25]

In Oct 1980 the CGPM decided that supplementary units were dimensionless derived units for which the CGPM immune the liberty of using them or not using them in expressions for SI derived units,^[43] on the basis that "[no formalism] exists which is at the same time coherent and convenient and in which the quantities plane angle and solid bending might be considered equally base of operations quantities" and that "[the possibility of treating the radian and steradian as SI base units] compromises the internal coherence of the SI based on only seven base of operations units".^[45] In 1995 the CGPM eliminated the class of supplementary units and defined the radian and the steradian equally "dimensionless derived units, the names and symbols of which may, but demand not, exist used in expressions for other SI derived units, as is convenient".^[46] Mikhail Kalinin writing in 2019 has criticized the 1980 CGPM conclusion every bit "unfounded" and says that the 1995 CGPM conclusion used inconsistent arguments and introduced "numerous discrepancies, inconsistencies, and contradictions in the wordings of the SI".^[47]

At the 2013 meeting of the CCU, Peter Mohr gave a presentation on alleged inconsistencies arising from defining the radian every bit a dimensionless unit of measurement rather than a base unit. CCU President Ian 1000. Mills declared this to be a "formidable trouble" and the CCU Working Grouping on Angles and Dimensionless Quantities in the SI was established.^[48] The CCU met nigh recently in 2021,^[update] but did not attain a consensus. A small number of members argued strongly that the radian should exist a base of operations unit, but the bulk felt the status quo was acceptable or that the change would cause more bug than it would solve. A task group was established to "review the historical use of SI supplementary units and consider whether reintroduction would be of benefit", among other activities.^{[49] [50]}

See also

• Angular frequency
• Minute and 2d of arc
• Steradian, a higher-dimensional analog of the radian which measures solid angle
• Trigonometry

Notes

References

1. ^ "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures. Archived from the original on 2018-12-25. Retrieved 2014-09-23 .
2. ^ International Agency of Weights and Measures 2019, p. 151: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, every bit dimensionless derived units."
3. ^ Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, p. APP-4, LCCN 76087042
4. ^ ^{a b c} International Agency of Weights and Measures 2019, p. 151.
5. ^ ^{a b} Weisstein, Eric W. "Radian". mathworld.wolfram.com . Retrieved 2020-08-31 .
6. ^ "ISO 80000-3:2006 Quantities and Units - Infinite and Fourth dimension".
7. ^ Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter VII. The Full general Bending [55] Signs and Limitations in Value. Practice XV.". Written at Ann Arbor, Michigan, The states. Trigonometry. Vol. Office I: Airplane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. South. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, Us. p. 73. Retrieved 2017-08-12 .
8. ^ International Agency of Weights and Measures 2019, p. 151: "One radian corresponds to the angle for which south = r"
9. ^ Quincey 2016, p. 844: "Also, as alluded to in Mohr & Phillips 2015, the radian can be divers in terms of the area A of a sector ( A = i/2 θ r ^two ), in which example it has the units thousand²⋅yard⁻²."
10. ^ International Agency of Weights and Measures 2019, p. 151: "One radian corresponds to the angle for which s = r, thus one rad = i."
11. ^ International Bureau of Weights and Measures 2019, p. 137.
12. ^ Bridgman, Percy Williams (1922). Dimensional analysis. New Haven : Yale University Printing. Angular aamplitude of swing [...] No dimensions.
13. ^ Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit". The Physics Teacher. 31 (2): 84–87. Bibcode:1993PhTea..31...84A. doi:10.1119/1.2343667.
14. ^ Prando, Giacomo (August 2020). "A spectral unit". Nature Physics. 16 (eight): 888. Bibcode:2020NatPh..16..888P. doi:10.1038/s41567-020-0997-iii. S2CID 225445454.
15. ^ Leonard, William J. (1999). Minds-on Physics: Advanced topics in mechanics. Kendall Hunt. p. 262. ISBN978-0-7872-5412-four.
16. ^ French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)". The Physics Teacher. xxx (5): 260–261. doi:10.1119/1.2343535.
17. ^ Oberhofer, Eastward. S. (March 1992). "What happens to the 'radians'?". The Physics Teacher. xxx (iii): 170–171. Bibcode:1992PhTea..30..170O. doi:x.1119/one.2343500.
18. ^ Brinsmade 1936; Romain 1962; Eder 1982; Torrens 1986; Brownstein 1997; Lévy-Leblond 1998; Foster 2010; Mills 2016; Quincey 2021; Leonard 2021; Mohr et al. 2022
19. ^ Mohr & Phillips 2015.
20. ^ ^{a b c d} Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base of operations quantity in the SI". Metrologia. 53 (three): 998–1002. arXiv:1604.02373. Bibcode:2016Metro..53..998Q. doi:10.1088/0026-1394/53/3/998. S2CID 119294905.
21. ^ ^{a b} Quincey 2016.
22. ^ ^{a b} Torrens 1986.
23. ^ Mohr et al. 2022, p. six.
24. ^ Mohr et al. 2022, pp. 8–9.
25. ^ ^{a b c d} Quincey 2021.
26. ^ Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems". Metrologia. 54 (4): 454–460. arXiv:1705.03765. Bibcode:2017Metro..54..454Q. doi:10.1088/1681-7575/aa7160. S2CID 119418270.
27. ^ Schabel, Matthias C.; Watanabe, Steven. "Boost.Units FAQ - ane.79.0". www.boost.org . Retrieved 5 May 2022. Angles are treated as units
28. ^ Mohr et al. 2022, p. 3.
29. ^ "UnityDimensions—Wolfram Language Documentation". reference.wolfram.com . Retrieved 1 July 2022.
30. ^ Luckey, Paul (1953) [Translation of 1424 book]. Siggel, A. (ed.). Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi [Treatise on the Circumference of al-Kashi]. Berlin: Akademie Verlag. p. 40.
31. ^ ^{a b} Roche, John J. (21 December 1998). The Mathematics of Measurement: A Critical History. Springer Science & Business Media. p. 134. ISBN978-0-387-91581-4.
32. ^ O'Connor, J. J.; Robertson, E. F. (Feb 2005). "Biography of Roger Cotes". The MacTutor History of Mathematics. Archived from the original on 2012-10-19. Retrieved 2006-04-21 .
33. ^ Cotes, Roger (1722). "Editoris notæ ad Harmoniam mensurarum". In Smith, Robert (ed.). Harmonia mensurarum (in Latin). Cambridge, England. pp. 94–95. In Canone Logarithmico exhibetur Systema quoddam menfurarum numeralium, quæ Logarithmi dicuntur: atque hujus systematis Modulus is est Logarithmus, qui metitur Rationem Modularem in Corol. vi. definitam. Similiter in Canone Trigonometrico finuum & tangentium, exhibetur Systema quoddam menfurarum numeralium, quæ Gradus appellantur: atque hujus systematis Modulus is est Numerus Graduum, qui metitur Angulum Modularem modo definitun, hoc est, qui continetur in arcu Radio æquali. Eft autem hic Numerus advertizement Gradus 180 ut Circuli Radius ad Semicircuinferentiam, hoc eft ut 1 ad 3.141592653589 &c. Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. Cujus Reciprocus eft 0.0174532925 &c. Hujus moduli subsidio (quem in chartula quadam Auctoris manu descriptum inveni) commodissime computabis mensuras angulares, queinadmodum oftendam in Nota Iii. [In the Logarithmic Canon in that location is presented a certain system of numerical measures called Logarithms: and the Modulus of this organisation 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 sure organization of numerical measures called Degrees: and the Modulus of this system is the Number of Degrees which measures the Modular Angle divers 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 i to 3.141592653589 &c. Hence the Modulus of the Trigonometric Canon will be 57.2957795130 &c. Whose Reciprocal is 0.0174532925 &c. With the aid of this modulus (which I constitute described in a note in the hand of the Author) you will nigh conveniently calculate the angular measures, every bit mentioned in Note III.]
34. ^ Gowing, Ronald (27 June 2002). Roger Cotes - Natural Philosopher. Cambridge Academy Press. ISBN978-0-521-52649-4.
35. ^ Euler, Leonhard. Theoria Motus Corporum Solidorum seu Rigidorum [Theory of the motion of solid or rigid bodies] (PDF) (in Latin). Translated by Bruce, Ian. Definition 6, paragraph 316.
36. ^ Isaac Todhunter, Aeroplane Trigonometry: For the Utilise of Colleges and Schools, p. ten, Cambridge and London: MacMillan, 1864 OCLC 500022958
37. ^ Cajori, Florian (1929). History of Mathematical Notations . Vol. 2. Dover Publications. pp. 147–148. ISBN0-486-67766-iv.
38. ^
    • Muir, Thos. (1910). "The Term "Radian" in Trigonometry". Nature. 83 (2110): 156. Bibcode:1910Natur..83..156M. doi:x.1038/083156a0. S2CID 3958702.
    • Thomson, James (1910). "The Term "Radian" in Trigonometry". Nature. 83 (2112): 217. Bibcode:1910Natur..83..217T. doi:10.1038/083217c0. S2CID 3980250.
    • Muir, Thos. (1910). "The Term "Radian" in Trigonometry". Nature. 83 (2120): 459–460. Bibcode:1910Natur..83..459M. doi:10.1038/083459d0. S2CID 3971449.
39. ^ Miller, Jeff (Nov 23, 2009). "Earliest Known Uses of Some of the Words of Mathematics". Retrieved Sep 30, 2011.
40. ^ Frederick Sparks, Longmans' School Trigonometry, p. 6, London: Longmans, Green, and Co., 1890 OCLC 877238863 (1891 edition)
41. ^ Quincey, Paul; Mohr, Peter J; Phillips, William D (1 Baronial 2019). "Angles are inherently neither length ratios nor dimensionless". Metrologia. 56 (four): 043001. arXiv:1909.08389. Bibcode:2019Metro..56d3001Q. doi:ten.1088/1681-7575/ab27d7. S2CID 198428043.
42. ^ Le Système international d'unités (PDF) (in French), 1970, p. 12, Pour quelques unités du Système International, la Conférence Générale north'a pas ou n'a pas encore décidé s'il due south'agit d'unités de base ou bien d'unités dérivées. [For some units of the SI, the CGPM yet hasn't yet decided whether they are base units or derived units.]
43. ^ ^{a b} Nelson, Robert A. (March 1984). "The supplementary units". The Physics Teacher. 22 (3): 188–193. Bibcode:1984PhTea..22..188N. doi:ten.1119/1.2341516.
44. ^ Report of the 7th meeting (PDF) (in French), Consultative Commission for Units, May 1980, pp. half-dozen–vii
45. ^ International Bureau of Weights and Measures 2019, pp. 174–175.
46. ^ International Bureau of Weights and Measures 2019, p. 179.
47. ^ Kalinin, Mikhail I (1 December 2019). "On the status of aeroplane and solid angles in the International System of Units (SI)". Metrologia. 56 (6): 065009. arXiv:1810.12057. Bibcode:2019Metro..56f5009K. doi:10.1088/1681-7575/ab3fbf. S2CID 53627142.
48. ^ Consultative Commission for Units (11–12 June 2013). Report of the 21st meeting to the International Committee for Weights and Measures (Written report). pp. 18–20.
49. ^ Consultative Committee for Units (21–23 September 2021). Study of the 25th meeting to the International Committee for Weights and Measures (Study). pp. 16–17.
50. ^ "CCU Job Group on angle and dimensionless quantities in the SI Brochure (CCU-TG-ADQSIB)". BIPM. Retrieved 26 June 2022.

• International Bureau of Weights and Measures (twenty May 2019), The International System of Units (SI) (PDF) (9th ed.), ISBN978-92-822-2272-0, archived (PDF) from the original on 8 May 2021
• Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly". American Journal of Physics. 4 (4): 175–179. Bibcode:1936AmJPh...iv..175B. doi:x.1119/1.1999110.
• Romain, Jacques Due east. (July 1962). "Angle equally a quaternary fundamental quantity". Journal of Research of the National Bureau of Standards, Section B. 66B (3): 97. doi:x.6028/jres.066B.012.
• Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"". Metrologia. eighteen (1): 1–12. Bibcode:1982Metro..xviii....1E. doi:10.1088/0026-1394/xviii/i/002. S2CID 250750831.
• Torrens, A B (i January 1986). "On Angles and Angular Quantities". Metrologia. 22 (1): ane–7. Bibcode:1986Metro..22....1T. doi:10.1088/0026-1394/22/1/002. S2CID 250801509.
• Brownstein, K. R. (July 1997). "Angles—Allow'due south treat them squarely". American Periodical of Physics. 65 (7): 605–614. Bibcode:1997AmJPh..65..605B. doi:ten.1119/1.18616.
• Lévy-Leblond, Jean-Marc (September 1998). "Dimensional angles and universal constants". American Periodical of Physics. 66 (9): 814–815. Bibcode:1998AmJPh..66..814L. doi:10.1119/i.18964.
• Foster, Marcus P (1 Dec 2010). "The side by side 50 years of the SI: a review of the opportunities for the eastward-Scientific discipline age". Metrologia. 47 (half dozen): R41–R51. doi:x.1088/0026-1394/47/half dozen/R01. S2CID 117711734.
• Mohr, Peter J; Phillips, William D (1 February 2015). "Dimensionless units in the SI". Metrologia. 52 (one): forty–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40.
• Quincey, Paul (i April 2016). "The range of options for handling plane angle and solid angle within a organization of units". Metrologia. 53 (2): 840–845. Bibcode:2016Metro..53..840Q. doi:x.1088/0026-1394/53/ii/840. S2CID 125438811.
• Mills, Ian (ane June 2016). "On the units radian and cycle for the quantity plane bending". Metrologia. 53 (iii): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991. S2CID 126032642.
• Quincey, Paul (1 Oct 2021). "Angles in the SI: a detailed proposal for solving the problem". Metrologia. 58 (five): 053002. arXiv:2108.05704. Bibcode:2021Metro..58e3002Q. doi:10.1088/1681-7575/ac023f. S2CID 236547235.
• Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent handling of angle and solid bending by the International System of Units (SI)". Metrologia. 58 (5): 052001. Bibcode:2021Metro..58e2001L. doi:ten.1088/1681-7575/abe0fc. S2CID 234036217.
• Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022). "On the dimension of angles and their units". Metrologia. 59 (5): 053001. arXiv:2203.12392. doi:ten.1088/1681-7575/ac7bc2.

External links

Look up radian in Wiktionary, the gratis dictionary.

• Media related to Radian at Wikimedia Commons

What Times What Equals 360,

Source: https://en.wikipedia.org/wiki/Radian

Posted by: mccollisteraloortat.blogspot.com

Share this post