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| {{Two other uses||millirads|Rad (unit)|other uses|Radian (disambiguation)}}
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| {{Infobox Unit
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| | name = Radian
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| | standard = [[SI derived unit]]
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| | quantity = [[Angle]]
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| | symbol = rad
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| | symbol2 = {{sup|c}}
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| | units1 = [[Degree (angle)|degrees]]
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| | inunits1 = ≈ 57.295°
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| }}
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| [[File:Circle radians.gif|thumb|right|300px|An arc of a [[circle]] with the same length as the [[radius]] of that circle corresponds to an ''angle'' of 1 radian. A full circle corresponds to an angle of 2[[pi|π]] radians.]]
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| The '''radian''' is the standard unit of angular measure, used in many areas of [[mathematics]]. An angle's measurement in radians is numerically equal to the length of a corresponding arc of a [[unit circle]], so one radian is just under 57.3 [[Degree (angle)|degrees]] (when the arc length is equal to the radius). The unit was formerly an [[SI supplementary unit]], but this category was abolished in 1995 and the radian is now considered an [[SI derived unit]]. The SI unit of [[solid angle]] measurement is the [[steradian]].
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| The radian is represented by the symbol '''rad''' ([[Unicode]]-encoded as {{unichar|33AD}}). An alternative symbol is the superscript letter c, for "<u>c</u>ircular measure"—but this is infrequently used as it can be easily mistaken for a [[degree symbol]] (°). So for example, a value of 1.2 radians could be written as 1.2 rad or 1.2{{sup|c}}.
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| == Definition ==
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| Radian describes the plane [[angle]] [[subtended]] by a circular [[Arc (geometry)|arc]] as the length of the arc divided by the [[radius]] of the arc. One radian is the [[angle]] [[subtended]] at the center of a [[circle]] by an [[Arc (geometry)|arc]] that is equal in length to the [[radius]] of the circle. More generally, the [[magnitude (mathematics)|magnitude]] in radians of such 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 enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, ''s'' = ''rθ''.
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| As the ratio of two lengths, the radian is a "[[pure number]]" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol [[°]] is used.
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| [[File:2pi-unrolled.gif|thumb|right|A complete revolution is 2π radians (shown here with a circle of radius one and thus [[circumference]] 2π).]]
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| 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 2[[pi|π]]''r'' /''r'', or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.
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| == History ==
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| The concept of radian measure, as opposed to the degree of an angle, is normally credited to [[Roger Cotes]] in 1714.<ref>{{cite web|url = http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html | title = Biography of Roger Cotes | work=The MacTutor History of Mathematics |date = February 2005 | last1= O'Connor|first1= J. J. |first2= E. F. |last2=Robertson}}</ref> He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure. The idea of measuring angles by the length of the arc was used already by other mathematicians. For example [[al-Kashi]] (c. 1400) used so-called ''diameter parts'' as units where one diameter part was {{sfrac|1|60}} radian and they also used sexagesimal subunits of the diameter part.<ref>{{cite book|first=Paul|last= Luckey|editor-first=A. |editor-last=Siggel|location=Berlin|publisher= Akademie Verlag| origyear=Translation of 1424 book|year=1953| title=Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi|trans_title=Treatise on the Circumference of al-Kashi| number=6|pages= 40}}</ref>
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| The term ''radian'' first appeared in print on 5 June 1873, in examination questions set by [[James Thomson (engineer)|James Thomson]] (brother of [[Lord Kelvin]]) at [[Queen's University Belfast|Queen's College]], [[Belfast]]. He used the term as early as 1871, while in 1869, [[Thomas Muir (mathematician)|Thomas Muir]], then of the [[University of St Andrews]], vacillated between ''rad'', ''radial'' and ''radian''. In 1874, Muir adopted ''radian'' after a consultation with James Thomson.<ref>{{cite book| author-link=Florian Cajori|first=Florian|last= Cajori| year=1929| title=History of Mathematical Notations| volume= 2|pages= 147–148| isbn=0-486-67766-4}}</ref><ref>{{cite journal| journal=Nature| year=1910| volume= 83| pages=156|doi=10.1038/083156a0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2110|bibcode = 1910Natur..83..156M }}{{cite journal| journal=Nature| year=1910| volume= 83| pages=217|doi=10.1038/083217c0| title=The Term "Radian" in Trigonometry| last1=Thomson| first1=James| issue=2112|bibcode = 1910Natur..83..217T }}{{cite journal| journal=Nature| year=1910| volume= 83| pages=459–460|doi=10.1038/083459d0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2120|bibcode = 1910Natur..83..459M }}</ref><ref>{{cite web|url=http://jeff560.tripod.com/r.html|date=Nov 23, 2009| accessdate=Sep 30, 2011|last=Miller|first=Jeff |title= Earliest Known Uses of Some of the Words of Mathematics}}</ref>
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| == Conversions ==
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| ===Conversion between radians and degrees===
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| [[File:Degree-Radian Conversion.svg|thumb|300px|A chart to convert between degrees and radians]]
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| As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π.
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| :<math> \text{angle in degrees} = \text{angle in radians} \cdot \frac {180^\circ} {\pi}</math>
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| For example:
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| :<math>1 \text{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ </math>
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| :<math>2.5 \text{ rad} = 2.5 \cdot \frac {180^\circ} {\pi} \approx 143.2394^\circ </math>
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| <br />
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| :<math>\frac {\pi} {3} \text{ rad} = \frac {\pi} {3} \cdot \frac {180^\circ} {\pi} = 60^\circ </math>
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| Conversely, to convert from degrees to radians, multiply by π/180.
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| :<math> \text{angle in radians} = \text{angle in degrees} \cdot \frac {\pi} {180^\circ}</math>
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| For example:
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| :<math>1^\circ = 1 \cdot \frac {\pi} {180^\circ} \approx 0.0175 \text{ rad}</math>
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| <math>23^\circ = 23 \cdot \frac {\pi} {180^\circ} \approx 0.4014 \text{ rad}</math>
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| Radians can be converted to turns by dividing the number of radians by 2π.
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| ====Radian to degree conversion derivation====
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| The length of circumference of a circle is given by <math>2\pi r</math>, where <math>r</math> is the radius of the circle.
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| So the following equivalent relation is true:
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| <math>360^\circ \iff 2\pi r</math>{{pad|4em}}[Since a <math>360^\circ</math> sweep is needed to draw a full circle]
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| By the definition of radian, a full circle represents:
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| :<math>\frac{2\pi r}{r} \text{ rad}</math>
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| :<math>= 2\pi \text{ rad}</math>
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| Combining both the above relations:
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| :<math>2\pi \text{ rad} = 360^\circ</math>
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| :<math>\Rrightarrow 1 \text{ rad} = \frac{360^\circ}{2\pi}</math>
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| :<math>\Rrightarrow 1 \text{ rad} = \frac{180^\circ}{\pi}</math>
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| ===Conversion between radians and grads===
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| <math>2\pi</math> radians are equal to one [[turn (geometry)|turn]], which is 400<sup>g</sup>. So, to convert from radians to [[Grad (angle)|grads]] multiply by <math>200/\pi</math>, and to convert from grads to radians multiply by <math>\pi/200</math>. For example,
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| :<math>1.2 \text{ rad} = 1.2 \cdot \frac {200^\text{g}} {\pi} \approx 76.3944^\text{g}</math>
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| :<math>50^\text{g} = 50 \cdot \frac {\pi} {200^\text{g}} \approx 0.7854 \text{ rad}</math>
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| ===Conversion of some common angles===
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| The table shows the conversion of some common angles.
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| {|class = wikitable style="text-align:center;"
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| ! Units !! colspan=13 | Values
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| |-
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| |style = "background:#f2f2f2; text-align:left;" | '''[[Turn (geometry)|Turns]]'''
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| |style = "width:3em;" | 0
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| |style = "width:3em;" | {{sfrac|1|24}}
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| |style = "width:3em;" | {{sfrac|1|12}}
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| |style = "width:3em;" | {{sfrac|1|10}}
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| |style = "width:3em;" | {{sfrac|1|8}}
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| |style = "width:3em;" | {{sfrac|1|6}}
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| |style = "width:3em;" | {{sfrac|1|5}}
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| |style = "width:3em;" | {{sfrac|1|4}}
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| |style = "width:3em;" | {{sfrac|1|3}}
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| |style = "width:3em;" | {{sfrac|2|5}}
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| |style = "width:3em;" | {{sfrac|1|2}}
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| |style = "width:3em;" | {{sfrac|3|4}}
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| |style = "width:3em;" | 1
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| |-
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| |style = "background:#f2f2f2; text-align:left;" | '''Radians'''
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| | 0
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| | {{sfrac|1|12}}π
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| | {{sfrac|1|6}}π
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| | {{sfrac|1|5}}π
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| | {{sfrac|1|4}}π
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| | {{sfrac|1|3}}π
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| | {{sfrac|2|5}}π
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| | {{sfrac|1|2}}π
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| | {{sfrac|2|3}}π
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| | {{sfrac|4|5}}π
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| | π
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| | {{sfrac|3|2}}π
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| | 2π
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| |-
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| |style = "background:#f2f2f2; text-align:left;" | '''[[Degree (angle)|Degrees]]'''
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| |style = "width:3em;" | 0°
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| |style = "width:3em;" | 15°
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| |style = "width:3em;" | 30°
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| |style = "width:3em;" | 36°
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| |style = "width:3em;" | 45°
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| |style = "width:3em;" | 60°
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| |style = "width:3em;" | 72°
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| |style = "width:3em;" | 90°
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| |style = "width:3em;" | 120°
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| |style = "width:3em;" | 144°
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| |style = "width:3em;" | 180°
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| |style = "width:3em;" | 270°
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| |style = "width:3em;" | 360°
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| |-
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| |style = "background:#f2f2f2; text-align:left;" | '''[[Grad (angle)|Grads]]'''
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| | 0<sup>g</sup>
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| | {{sfrac|16|2|3}}<sup>g</sup>
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| | {{sfrac|33|1|3}}<sup>g</sup>
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| | 40<sup>g</sup>
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| | 50<sup>g</sup>
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| | {{sfrac|66|2|3}}<sup>g</sup>
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| | 80<sup>g</sup>
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| | 100<sup>g</sup>
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| | {{sfrac|133|1|3}}<sup>g</sup>
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| | 160<sup>g</sup>
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| | 200<sup>g</sup>
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| | 300<sup>g</sup>
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| | 400<sup>g</sup>
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| |}
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| ==Advantages of measuring in radians==
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| [[File:Radian-common.svg|thumb|357px|right|Some common angles, measured in radians. All the polygons are regular polygons.]]
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| 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.
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| Most notably, results in [[analysis (mathematics)|analysis]] involving [[trigonometric function]]s are simple and elegant when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple [[limit of a function|limit]] formula
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| :<math>\lim_{h\rightarrow 0}\frac{\sin h}{h}=1,</math>
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| which is the basis of many other identities in mathematics, including
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| :<math>\frac{d}{dx} \sin x = \cos x</math>
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| :<math>\frac{d^2}{dx^2} \sin x = -\sin x.</math>
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| 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 <math> \frac{d^2 y}{dx^2} = -y </math>, the evaluation of the integral <math> \int \frac{dx}{1+x^2} </math>, 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.
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| The trigonometric functions also have simple and elegant series expansions when radians are used; for example, the following [[Taylor series]] for sin ''x'' :
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| :<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots .</math>
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| 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'' = π''x'' /180, so
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| :<math>\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 .</math>
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| Mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) are, again, elegant when the functions' arguments are in radians and messy otherwise.
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| == Dimensional analysis ==
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| Although the radian is a unit of measure, it is a [[Dimensionless number|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.
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| Although [[Polar coordinates|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.<ref>For a debate on this meaning and use see:
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| {{cite journal|doi=10.1119/1.18616|title=Angles—Let's treat them squarely|year=1997|last1=Brownstein|first1=K. R.|journal=American Journal of Physics|volume=65|issue=7|pages=605|bibcode = 1997AmJPh..65..605B }},
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| {{cite journal|title=Angles as a fourth fundamental quantity|year=1962|last1=Romain|first1=J.E.|journal=Journal of Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics|volume=66B|issue=3|pages=97}},
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| {{cite journal|doi=10.1119/1.18964|title=Dimensional angles and universal constants|year=1998|last1=LéVy-Leblond|first1=Jean-Marc|journal=American Journal of Physics|volume=66|issue=9|pages=814|bibcode = 1998AmJPh..66..814L }}, and {{cite journal|doi=10.1119/1.19185|title=Units—SI-Only, or Multicultural Diversity?|year=1999|last1=Romer|first1=Robert H.|journal=American Journal of Physics|volume=67|pages=13 |bibcode = 1999AmJPh..67...13R }}</ref>
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| ==Use in physics==
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| 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.
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| Similarly, [[angular acceleration]] is often measured in radians per second per second (rad/s<sup>2</sup>).
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| For the purpose of dimensional analysis, the units are s<sup>−1</sup> and s<sup>−2</sup> respectively.
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| 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 (waves)|phase]], whilst if the phase difference of two waves is (k·2π + π), where k is an integer, they are considered in antiphase.
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| == Multiples of radian units ==
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| [[Metric prefix]]es have limited use with radians, and none in mathematics.
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| There are 2[[π]] × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a trigonometric milliradian is just under {{frac|1|6283}} of a circle. This “real” trigonometric unit of angular measurement of a circle is in use by [[telescopic sight]] manufacturers using [[Stadiametric rangefinding|(stadiametric) rangefinding]] in [[reticle]]s.
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| The [[beam divergence|divergence]] of [[laser]] beams is also usually measured in milliradians.
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| An approximation of the trigonometric milliradian (0.001 rad), known as the [[Angular mil|(angular) mil]], is used by [[NATO]] and other military organizations in [[gun]]nery and [[Sniper#Targeting|targeting]]. Each angular mil represents {{frac|1|6400}} of a circle and is 1-⅞% smaller than the trigonometric 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 {{frac|1|2000π}}; for example Sweden used the {{frac|1|6300}} ''streck'' and the USSR used {{frac|1|6000}}.
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| 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).
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| 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. Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.
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| == See also ==
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| * [[Angular frequency]]
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| * [[Angular mil]] - military measurement
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| * [[Grad (angle)|Grad]]
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| * [[Harmonic analysis]]
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| * [[Steradian]] - the "square radian"
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| * [[Trigonometry]]
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| == Notes and references ==
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| {{Reflist|2}}
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| == External links ==
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| {{Wikibooks|Trigonometry/Radian and degree measures}}
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| {{Wiktionary|radian}}
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| * [http://mathworld.wolfram.com/Radian.html Radian] at [[MathWorld]]
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| {{SI units}}
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| [[Category:Natural units]]
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| [[Category:SI derived units]]
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| [[Category:Pi]]
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| [[Category:Units of angle]]
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