# Dictionary Definition

radian n : the unit of plane angle adopted under
the Systeme International d'Unites; equal to the angle at the
center of a circle subtended by an arc equal in length to the
radius (approximately 57.295 degrees) [syn: rad]

# User Contributed Dictionary

## English

### Etymology

From radi(us) + -an### Noun

- In the International System of Units, the derived unit of plane angular measure of angle equal to the angle subtended at the centre of a circle by an arc of its circumference equal in length to the radius of the circle. Symbol: rad

#### Derived terms

#### Translations

unit

## French

### Noun

radian m- radian

## Polish

### Noun

radian m- radian

## Swedish

### Noun

radian c- radian

# Extensive Definition

The radian is a unit of plane angle, equal to 180/π degrees,
or about 57.2958 degrees. It is the standard unit of angular
measurement in all areas of mathematics beyond the
elementary level.

The radian is represented by the symbol "rad" or,
more rarely, by the superscript c (for "circular measure"). For
example, an angle of 1.2 radians would be written as "1.2 rad" or
"1.2c" (the second symbol can be mistaken for a degree: "1.2°").
However, the radian is mathematically considered 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.

The radian 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.

## Definition

One radian is the angle subtended at the center of a
circle by an arc that
is equal in length to the radius of the circle.

More generally, the magnitude in radians of any
angle subtended by two radii is equal to the ratio of the length of
the enclosed arc 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.

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/π degrees.

## History

The concept of radian measure, as opposed to the
degree of an angle, should probably be credited to Roger Cotes
in 1714. He had the radian in everything but name, and he
recognized its naturalness as a unit of angular measure.

The term radian first appeared in print on
June 5,
1873, in
examination questions set by James
Thomson (brother of Lord Kelvin)
at Queen's
College, Belfast. He used
the term as early as 1871, while in 1869, 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.

## Conversions

### Conversion between radians and degrees

As stated above, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,- 1 \mbox = 1 \cdot \frac \approx 57.2958^\circ
- 2.5 \mbox = 2.5 \cdot \frac \approx 143.2394^\circ
- \frac \mbox = \frac \cdot \frac = 60^\circ

Conversely, to convert from degrees to radians,
multiply by π/180. For example,

- 1^\circ = 1 \cdot \frac \approx 0.0175 \mbox
- 23^\circ = 23 \cdot \frac \approx 0.4014 \mbox

You can also convert radians to revolutions by
dividing number of radians by 2π.

The table shows the conversion of some common
angles.

### Conversion between radians and grads

2π radians are equal to one complete revolution,
which is 400g. So, to convert from radians to grads
multiply by 200/π, and to convert from grads to radians multiply by
π/200. For example,

- 1.2 \mbox = 1.2 \cdot \frac \approx 76.3944^
- 50^ = 50 \cdot \frac \approx 0.7854 \mbox

## Reasons why radians are preferred in mathematics

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 are simple and elegant when the functions' arguments
are expressed in radians. For example, the use of radians leads to
the simple limit
formula

- \lim_\frac=1,

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

- \frac \sin x = \cos x
- \frac \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 d2y/dx2 = −y, the
evaluation of the integral ∫dx/(1 + x2), 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, the
following Taylor
series for sin x :

- \sin x = x - \frac + \frac - \frac + \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

- \sin x\ (deg) = \sin y\ (rad) = \frac x - \left (\frac \right )^3\ \frac + \left (\frac \right )^5\ \frac - \left (\frac \right )^7\ \frac + \cdots .

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.

## 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
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.

Another way to see the dimensionlessness of the
radian is in the series representations of the trigonometric
functions, such as the Taylor
series for sin x mentioned earlier:

- \sin x = x - \frac + \frac - \frac + \cdots .

If x had units, then the sum would be
meaningless: the linear term x cannot be added to (or have
subtracted) the cubic term x^3/3! or the quintic term x^5/5!, etc.
Therefore, x must be dimensionless.

## 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).

The reasons are the same as in mathematics.

## Multiples of radian units

Metric prefixes
have limited use with radians, and none in mathematics.

The milliradian (0.001 rad, or 1 mrad) is used in
gunnery and targeting,
because it corresponds to an error of 1 m at a range of 1000 m (at
such small angles, the curvature is negligible). The divergence
of laser beams is also
usually measured in milliradians.

Smaller units like microradians (μrads) and
nanoradians (nrads) 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.

However, the larger prefixes have no apparent
utility, mainly because to exceed 2π radians is to begin the same
circle (or revolutionary cycle) again.

## See also

- Angular mil - military measurement
- Trigonometry
- Harmonic analysis
- Angular frequency
- Grad
- Degree
- Steradian - the "square radian"

radian in Arabic: راديان

radian in Afrikaans: Radiaal

radian in Bulgarian: Радиан

radian in Catalan: Radiant (angle)

radian in Czech: Radián

radian in Danish: Radian

radian in German: Radiant (Einheit)

radian in Estonian: Radiaan

radian in Spanish: Radián

radian in Esperanto: Radiano

radian in Basque: Radian

radian in French: Radian

radian in Galician: Radián

radian in Korean: 라디안

radian in Croatian: Radijan

radian in Indonesian: Radian

radian in Italian: Radiante

radian in Hebrew: רדיאן

radian in Malay (macrolanguage): Radian

radian in Dutch: Radiaal

radian in Japanese: ラジアン

radian in Norwegian: Radian

radian in Norwegian Nynorsk: Radian

radian in Polish: Radian

radian in Portuguese: Radiano

radian in Romanian: Radian

radian in Russian: Радиан

radian in Simple English: Radian

radian in Slovak: Radián

radian in Slovenian: Radian

radian in Serbian: Радијан

radian in Finnish: Radiaani

radian in Swedish: Radian

radian in Thai: เรเดียน

radian in Ukrainian: Радіан

radian in Chinese: 弧度