radian

Dictionary: ra·di·an (rā'dē-ən) pronunciation

n. (Abbr. rad)
A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, approximately 57°17′44.6″.

[RADI(US) + -AN¹.]

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Statistics Dictionary: radian

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Variant: rad

A unit of angle measurement. The radian measure of the angle enclosed between two lines OP and OQ is defined to be the length of the circular arc, with centre O and unit radius, enclosed between the lines. Thus a right angle is ½π rad, and a complete revolution corresponds to 2π rad. The relationship with degrees is that π rad=180° or 1 rad≈57.296°. Use of radians is important when trigonometrical functions are used in calculus. For example in the relationship that the derivative of sin x is cos x.

US Military Dictionary: rad

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n.a unit of absorbed dose of ionizing radiation.

See the Introduction, Abbreviations and Pronunciation for further details.

Measures and Units: radian

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plane angle. Symbol rad. (Metric) The angle subtended by a section of circumference equal in length to the radius (hence slightly less than the 60° angle subtended by the secant of the same length that forms a side of the readily constructed hexagon). Quantitatively the ratio of the circumferential length to the radial length. Since the circumference of the complete circle is 2π times the radius, a complete revolution equals 2π rad = 6.283 185~ rad; 1 rad = ^360°/_2π = 57.295 78~ °. Hence

• rad·s^-1 for angular frequency, angular speed, revolution speed;
• rad·s^-2 for angular acceleration.

Creating an equivalent base unit in the SI, giving dimensionality to the plane angle, has been proposed.
[Eder W. E. Metrologia Vol. 18, 1-12 (1982)]
[Eder W. E. Metrologia Vol. 19, 1-8 (1983)]
[Torrens A. B. Metrologia Vol. 22, 1-7 (1986)]
[Torrens A. B. Metrologia Vol. 23, 57-8 (1986)] (See also spat.) But the radian of the SI, once a supplementary unit, since 1980 is a dimensionless derived unit.
[Giacomo P. Metrologia Vol. 17, 69-74 (1981)]

With the irrational quantity 2π per complete turn or revolution, the radian is awkward in various ways. It has influenced the setting of conventional values for units (see permittivity). While the degree is preferable for most working purposes, the radian is the natural unit in any mathematical equations, e.g. for map projecting despite the degree being the convenient expression on the map.

The radian is sometimes divided into 100 centrads, and into 1 000 mils.

1960	11th CGPM
1980	CIPM: ‘considering
	— that the units radian and steradian are usually introduced into expressions for units when there is need for clarification, especially in photometry where the steradian plays an important role in distinguishing between units corresponding to different quantities,
	— that in the equations used one generally expresses plane angle as the ratio of two lengths and the solid angle as the ratio between an area and the square of a length, and consequently that these quantities are treated as dimensionless quantities,
	— that the study of formalisms in use in the scientific field shows that none exists which is at the same time coherent and convenient and in which the quantities plane angle and solid angle might be considered as base quantities,
	considering also
	— that the interpretation given by the CIPM in 1969 for the class of supplementary units introduced in Resolution 12 of the 11th CGPM in 1960 allows the freedom of treating the radian and the steradian as SI base units,
	— that such a possibility compromises the internal coherence of the SI based on only seven base units, decides to interpret the class of supplementary units in the International System as a class of dimensionless derived units for which the CGPM allows the freedom of using or not using them in expressions for SI derived units.’see note below

[Le Système International d'Unités (Sèvres, France: Bureau International de Poids et Mesures, 1985)]

Sports Science and Medicine: radian

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rad

A unit of angular measure. The radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. 1 rad = 57.3°. Radians are often quantified in multiples of pi. One complete circle (equal to 360°) is an arc of 2 pi radians.

Radian

Veterinary Dictionary: rad

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1. acronym for radiation absorbed dose; a superseded, non-SI unit of measurement of the absorbed dose of ionizing radiation. It corresponds to an energy transfer of 100 ergs per gram of any absorbing material (including tissue). The biological effect of 1 rad of radiation varies with the type of radiation. When the dose is in rem, all types have the same biological effect. Now replaced by the gray.
2. abbreviation for radiograph; used in medical records.

Unit Conversions: radians

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To convert from radians to:

degrees, multiply by 57.29578.
minutes, multiply by 3438.
seconds, multiply by 2.063E+05.

Related measurements:
radians/sec
radians/sec/sec

Wikipedia: Radian

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An angle of 1 radian results in an arc with an equal length to the radius of the circle.

For the musical group, see Radian (band). For the comic book character, see Radian (comics).

Also, mrad redirects here; for millirads see Rad (unit).

The radian is a unit of plane angle, equal to 180/π (or 360/(2π)) degrees, or about 57.2958 degrees, or about 57°17′45″. 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 " 1.2^c " (the second symbol is often 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.

1 Definition
2 History
3 Conversions
- 3.1 Conversion between radians and degrees
- 3.2 Conversion between radians and grads
4 Advantages of measuring in radians
5 Dimensional analysis
6 Use in physics
7 Multiples of radian units
8 See also
9 References
10 External links

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 an angle subtended by a given length s is equal to the ratio of s 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.^[1] 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 5 June 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.^[2]^[3]^[4]

Conversions

Conversion between radians and degrees

A chart to convert between degrees and radians

As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,

$1 \mbox{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ$

$2.5 \mbox{ rad} = 2.5 \cdot \frac {180^\circ} {\pi} \approx 143.2394^\circ$

$\frac {\pi} {3} \mbox{ rad} = \frac {\pi} {3} \cdot \frac {180^\circ} {\pi} = 60^\circ$

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

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

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

Radians can be converted to revolutions by dividing the number of radians by 2π.

Conversion between radians and grads

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

$1.2 \mbox{ rad} = 1.2 \cdot \frac {200^{\rm g}} {\pi} \approx 76.3944^{\rm g}$

$50^{\rm g} = 50 \cdot \frac {\pi} {200^{\rm g}} \approx 0.7854 \mbox{ rad}$

The table shows the conversion of some common angles.

Units	Values
Revolutions	0	1/12	1/8	1/6	1/4	1/2	3/4	1
Degrees	0°	30°	45°	60°	90°	180°	270°	360°
Radians	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\pi\,$	$\frac{3\pi}{2}$	$2\pi\,$
Grads	0^g	$\frac{100}{3}^{\rm g}$	50^g	$\frac{200^{\rm g}}{3}$	100^g	200^g	300^g	400^g

Advantages of measuring in radians

Some common angles, measured in radians. All the polygons 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 are simple and elegant when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

$\lim_{h\rightarrow 0}\frac{\sin h}{h}=1,$

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

$\frac{d}{dx} \sin x = \cos x$

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

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 $\frac{d^2 y}{dx^2} = -y$ , the evaluation of the integral $\int \frac{dx}{1+x^2}$ , 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, the following Taylor series for sin x :

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

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

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

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 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.

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:

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

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.

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.

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 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 2π radians, they are considered in phase, whilst if the phase difference of two waves is π, they are considered in antiphase.

Multiples of radian units

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

An approximation of the milliradian (0.001 rad), known as the Mil is used in gunnery and targeting. Based upon an approximation of π=3.2, there are 6400 milliradians in a complete rotation. Other gunnery systems may use a different approximation to π. Being based on the milliradian, it corresponds roughly 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.

References

^ O'Connor, J.J. and E.F. Robertson (February 2005). "Biography of Roger Cotes". The MacTutor History of Mathematics. http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html.
^ Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147–148
^ Nature, 1910, Vol. 83, pp. 156, 217, and 459–460
^ Earliest Known Uses of Some of the Words of Mathematics

External links

Wikibooks has a book on the topic of

Trigonometry/Radian and degree measures

Look up radian in Wiktionary, the free dictionary.

Radian at MathWorld

v • d • e SI derived units

Derived units	radian · steradian · hertz · newton · pascal · joule · watt · coulomb · volt · farad · ohm · henry · siemens · weber · tesla · Celsius · lumen · lux · becquerel · gray · sievert · katal

Base units	metre · kilogram · second · ampere · kelvin · candela · mole

SI derived units

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