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radian

  (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.]


 
 
US Military Dictionary: rad

n.a unit of absorbed dose of ionizing radiation.

See the Introduction, Abbreviations and Pronunciation for further details.

 
Measures and Units: radian

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.

196011th CGPM
1980CIPM: ‘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

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
Radian

 
Veterinary Dictionary: rad

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

To convert from radians to:

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

Convert:  Into: 
Result: 
Related measurements:
radians/sec
radians/sec/sec


 
Wikipedia: radian
For the musical group, see Radian (band). For the comic book character, see Radian (comics).
Some common angles, measured in radians. All the polygons are regular polygons.
Enlarge
Some common angles, measured in radians. All the polygons are regular polygons.

The radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It 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 the de facto unit of angular measurement for mathematicians, 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

An angle of 1 radian subtends an arc equal in length to the radius of the circle.
Enlarge
An angle of 1 radian subtends an arc equal in length to the radius of the circle.

One radian is the angle subtended at the center of a circle by an arc of length equal 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 a 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 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.[2][3][4]

Conversions

Conversion between radians and degrees

As explained above under "Definition", 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 \times \frac {180^\circ} {\pi} \approx 57.2958^\circ
2.5 \mbox{ rad} = 2.5 \times \frac {180^\circ} {\pi} \approx 143.2394^\circ
\frac {\pi} {3} \mbox{ rad} = \frac {\pi} {3} \times \frac {180^\circ} {\pi} = 60^\circ

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

1^\circ = 1 \times \frac {\pi} {180^\circ} \approx 0.0175 \mbox{ rad}
23^\circ = 23 \times \frac {\pi} {180^\circ} \approx 0.4014 \mbox{ rad}

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

The table shows the conversion of some common angles.

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\,

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{ rad} = 1.2 \times \frac {200^g} {\pi} \approx 76.3944^g
50^g = 50 \times \frac {\pi} {200^g} \approx 0.7854 \mbox{ rad}

Reasons why radians are preferred in mathematics

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. One important reason is that results involving trigonometric functions are simple and "natural" when the function's argument is expressed in radians. For example, the use of radians leads to the simple identity

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

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

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

The trigonometric functions also have simpler 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.

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{x^3}{3!} + \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 x3 / 3!. 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).

SI multiples

SI prefixes have limited use with radians. The milliradian (0.001 rad, or 1 mrad) is used in gunnery and general targeting, because it corresponds to 1 m at a range of 1000 m (at such small angles, the curvature can be considered 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

References

  1. ^ Biography of Roger Cotes, The MacTutor History of Mathematics
  2. ^ Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147–148
  3. ^ Nature, 1910, Vol. 83, pp. 156, 217, and 459–460
  4. ^ Earliest Known Uses of Some of the Words of Mathematics

External links


This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

 
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Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
US Military Dictionary. The Oxford Essential Dictionary of the U.S. Military. Copyright © 2001, 2002 by Oxford University Press, Inc. All rights reserved.  Read more
Measures and Units. A Dictionary of Weights, Measures, and Units. Copyright © Donald Fenna 2002, 2004. All rights reserved.  Read more
Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved.  Read more
Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved.  Read more
Answers Corporation Unit Conversions. © 1999-2008 by Answers Corporation. All rights reserved.  Read more
Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Radian".  Read more

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