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Unit systemSI derived unit
Unit ofAngle
In unitsDimensionless with an arc length equal to the radius, i.e. 1 m/m
Conversions
1 rad in ...... is equal to ...
turns   1/2π turn
degrees   180/π ≈ 57.296°
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.

The radian, denoted by the symbol ${\displaystyle {\text{rad}}}$, is the SI unit for measuring angles, 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) and the radian is now an SI derived unit.[1] The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.

## Definition

One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle.[2] More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the intercepted arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = .

As the ratio of two lengths, the radian is a pure number.[a] In SI, the radian is defined as having the value 1.[6] As a consequence, in mathematical writing, the symbol "rad" is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used.

A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π).

It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π57.295779513082320876 degrees.[7]

The relation 2π rad = 360° can be derived using the formula for arc length, ${\displaystyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}$, and by using a circle of radius 1. Since radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, ${\displaystyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}$. This can be further simplified to ${\displaystyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}$. Multiplying both sides by 360° gives 360° = 2π rad.

## History

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714.[8][9] He described the radian in everything but name, and recognized its naturalness as a unit of angular measure. Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.[10]

The idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi (c. 1400) used so-called diameter parts as units, where one diameter part was 1/60 radian. They also used sexagesimal subunits of the diameter part.[11]

The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between the terms rad, radial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian.[12][13][14] The name radian was not universally adopted for some time after this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.[15]

## Unit symbol

The International Bureau of Weights and Measures[16] and International Organization for Standardization[17] specify rad as the symbol for the radian. Alternative symbols used 100 years ago are c (the superscript letter c, for "circular measure"), the letter r, or a superscript R,[18] but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence a value of 1.2 radians would most commonly be written as 1.2 rad; other notations include 1.2 r, 1.2rad, 1.2c, or 1.2R.

## Conversions

A chart to convert between degrees and radians
Conversion of common angles
1/2π turn 1 rad c. 57.3° c. 63.7g
1. ^ While the radian is normally defined as the ratio of two lengths (it is a "pure number"), Mohr and Phillips[3] and others [4] [5] point out that problems can arise if angles are defined to be dimensionless.
2. ^ In this table, 𝜏 denotes 2π.

### Conversion between radians and degrees

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

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

For example:

${\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$
${\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}$
${\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}$

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

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

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 be converted to turns (complete revolutions) by dividing the number of radians by 2π.

#### Radian to degree conversion derivation

The length of circumference of a circle is given by ${\displaystyle 2\pi r}$, where ${\displaystyle r}$ is the radius of the circle.

So the following equivalent relation is true:

${\displaystyle 360^{\circ }\iff 2\pi r}$ [Since a ${\displaystyle 360^{\circ }}$ sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

${\displaystyle {\frac {2\pi r}{r}}{\text{ rad}}}$
${\displaystyle =2\pi {\text{ rad}}}$

Combining both the above relations:

${\displaystyle 2\pi {\text{ rad}}=360^{\circ }}$
${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {360^{\circ }}{2\pi }}}$
${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {180^{\circ }}{\pi }}}$

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

${\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}$
${\displaystyle 50^{\text{g}}=50^{\text{g}}\cdot {\frac {\pi }{200^{\text{g}}}}\approx 0.7854{\text{ rad}}}$

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

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

Most 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

${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$[7]
${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}$

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 ${\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}$ and so on). In all such cases, it 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 have simple and elegant series expansions when radians are used. For example, when x is in radians, the Taylor series for sin x becomes:

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

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

${\displaystyle \sin x_{\mathrm {deg} }=\sin y_{\mathrm {rad} }={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}$

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

## Dimensional analysis

Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.

Although polar and spherical coordinates use radians to describe coordinates in two and three dimensions, the unit is derived from the radius coordinate, so the angle measure is still dimensionless.[19]

## Use in physics

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

Similarly, angular acceleration is often measured in radians per second per second (rad/s2).

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

Likewise, the phase difference of two waves can also be measured in radians. For example, if the phase difference of two waves is (k⋅2π) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is (k⋅2π + π), where k is an integer, they are considered in antiphase.

## SI multiples

There are 2π × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under 1/6283 of the angle subtended by a full circle. This "real" unit of angular measurement of a circle is in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

An approximation of the milliradian (0.001 rad) is used by 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 it introduces. In the past, other gunnery systems have used different approximations to 1/2000π; for example Sweden used the 1/6300 streck and the USSR used 1/6000. Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Smaller units like microradians (μrad) and nanoradians (nrad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is arc second, which is π/648,000 rad (around 4.8481 microradians). Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

## 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. ^ Protter, Murray H.; Morrey, Charles B., Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, p. APP-4, LCCN 76087042
3. ^ Mohr, J. C.; Phillips, W. D. (2015). "Dimensionless Units in the SI". Metrologia. 52 (1): 40–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40. S2CID 3328342.
4. ^ Mills, I. M. (2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991.
5. ^ "SI units need reform to avoid confusion". Editorial. Nature. 548 (7666): 135. 7 August 2011. doi:10.1038/548135b. PMID 28796224.
6. ^ ISO 80000-3:2006
7. ^ a b Weisstein, Eric W. "Radian". mathworld.wolfram.com. Retrieved 2020-08-31.
8. ^ O'Connor, J. J.; Robertson, E. F. (February 2005). "Biography of Roger Cotes". The MacTutor History of Mathematics. Archived from the original on 2012-09-24. Retrieved 2006-04-21.
9. ^ Roger Cotes died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, Harmonia mensurarum … . In a chapter of editorial comments by Smith, he gives, for the first time, the value of one radian in degrees. See: Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: Editoris notæ ad Harmoniam mensurarum, top of page 95. From page 95: After stating that 180° corresponds to a length of π (3.14159…) along a unit circle (i.e., π radians), Smith writes: "Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. " (Whence the unit of trigonometric measure, 57.2957795130… [degrees per radian], will appear.)
10. ^ Isaac Todhunter, Plane Trigonometry: For the Use of Colleges and Schools, p. 10, Cambridge and London: MacMillan, 1864 OCLC 500022958
11. ^ 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.
12. ^ Cajori, Florian (1929). History of Mathematical Notations. 2. Dover Publications. pp. 147–148. ISBN 0-486-67766-4.
13. ^ Muir, Thos. (1910). "The Term "Radian" in Trigonometry". Nature. 83 (2110): 156. Bibcode:1910Natur..83..156M. doi:10.1038/083156a0. 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.
14. ^ Miller, Jeff (Nov 23, 2009). "Earliest Known Uses of Some of the Words of Mathematics". Retrieved Sep 30, 2011.
15. ^ Frederick Sparks, Longmans' School Trigonometry, p. 6, London: Longmans, Green, and Co., 1890 OCLC 877238863 (1891 edition)
16. ^ 2019 BIPM Brochure
17. ^ ISO 80000-3:2006 Quantities and Units - Space and Time
18. ^ Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter VII. The General Angle [55] Signs and Limitations in Value. Exercise XV.". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 73. Retrieved 2017-08-12.
19. ^ For a debate on this meaning and use see: Brownstein, K. R. (1997). "Angles—Let's treat them squarely". American Journal of Physics. 65 (7): 605–614. Bibcode:1997AmJPh..65..605B. doi:10.1119/1.18616., Romain, J.E. (1962). "Angles as a fourth fundamental quantity". Journal of Research of the National Bureau of Standards Section B. 66B (3): 97. doi:10.6028/jres.066B.012., LéVy-Leblond, Jean-Marc (1998). "Dimensional angles and universal constants". American Journal of Physics. 66 (9): 814–815. Bibcode:1998AmJPh..66..814L. doi:10.1119/1.18964., and Romer, Robert H. (1999). "Units—SI-Only, or Multicultural Diversity?". American Journal of Physics. 67 (1): 13–16. Bibcode:1999AmJPh..67...13R. doi:10.1119/1.19185.
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