NationMaster - Encyclopedia: Radian

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. Radian is an Austrian expe, post-rock, jazz band that sounds like an electronic act. ... See Radian for the mathematical concept, or Radian (band) for the musical group Radian is a fictional mutant character in the Marvel Comics Universe. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... This article is about angles in geometry. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... This article describes the unit of angle. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...

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 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 degree symbol (Â°) is a typographical symbol, or glyph, that is used to represent degrees of arc (see Geographic coordinate system ) or temperature. ...

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. Until 1995, SI (International System of Units) supplementary units were: As of October 1995, the category of supplementary units has been abolished from the SI system of measurement, and the radian and the steradian are now considered SI derived units. ... SI derived units are part of the SI system of measurement units and are derived from the seven SI base units. ... A solid angle is the three dimensional analog of the ordinary angle. ... The steradian (ste from Greek stereos, solid) is the SI derived unit of solid angle, and the 3-dimensional equivalent of the radian. ...

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

Definition

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 circumference that is equal in length to the radius of the circle. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... This article is about an authentication, authorization, and accounting protocol. ... Circle illustration This article is about the shape and mathematical concept of circle. ... This article is about angles in geometry. ... In mathematics the term subtended usually refers to the direct relationship between an angle and its arc length. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... The circumference is the distance around a closed curve. ... This article is about an authentication, authorization, and accounting protocol. ...

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.^[1] He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure. Roger Cotes (Burbage, Leicestershire July 10, 1682 â€“ June 5, 1716 in Cambridge, Cambridgeshire) was an English mathematician. ...

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] is the 156th day of the year (157th in leap years) in the Gregorian calendar. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... James Thomson (February 16, 1822 - May 8, 1892) was an Irish engineer and physicist whose reputation would have been substantial had it not been overshadowed by that of his brother William Thomson, 1st Baron Kelvin. ... William Thomson, Archbishop of York, has the same name as this man. ... Queens University Belfast is a university in Belfast, Northern Ireland and a member of the Russell Group (a lobby group of major research universities in the United Kingdom). ... This article is about the city in Northern Ireland. ... There have been two notable Thomas Muirs: Thomas Muir, leader of the Scottish Friends of the People Society; and Thomas Muir, Scottish mathematician This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... St Marys College Bute Medical School St Leonards College[5][6] Affiliations 1994 Group Website http://www. ... Year 1874 (MDCCCLXXIV) was a common year starting on Thursday (link with display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ...

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,

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

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

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

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

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

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

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	$frac{pi}{6}$	$frac{pi}{4}$	$frac{pi}{3}$	$frac{pi}{2}$	$pi,$	$frac{3pi}{2}$	$2pi,$

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, The grad is a measurement of plane angles of value 1/400 of a full circle, thus dividing a right angle in 100. ...

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

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

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. For other uses, see Calculus (disambiguation). ...

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 Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... Sine redirects here. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

$lim_{hrightarrow 0}frac{sin h}{h}=1$ ,

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 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 : Series expansion redirects here. ...

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

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. The exponential function is one of the most important functions in mathematics. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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. In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ...

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: Series expansion redirects here. ...

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

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.

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. Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

Similarly, angular acceleration is often measured in radians per second per second (rad/s²). Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non...

The reasons are the same as in mathematics.

Multiples of radian units

Metric prefixes have limited use with radians, and none in mathematics. An SI prefix (also known as a metric prefix) is a name or associated symbol that precedes a unit of measure (or its symbol) to form a decimal multiple or submultiple. ...

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. The Gunnery is a coeducational college preparatory boarding and day school for 9th-12th grade students. ... For other uses, see Sniper (disambiguation). ... The beam divergence of an electromagnetic beam is the increase in beam diameter with distance from the aperture from which the beam emerges in any plane that intersects the beam axis. ... For other uses, see Laser (disambiguation). ...

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.

References

^ O'Connor, J.J. and E.F. Robertson (February 2005). Biography of Roger Cotes. The MacTutor History of Mathematics.
^ 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
Radians/Degree Converter[[af:Radiaal]

Categories: Natural units | SI derived units | Trigonometry | Units of angle

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

Radian - Wikipedia, the free encyclopedia (594 words)
	The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle is one radian.
	An angle measuring 1 radian subtends an arc equal in length to the radius of the circle.
	Although the radian is a unit of measure, anything measured in radians is dimensionless.

Radian - definition of Radian in Encyclopedia (281 words)
	In mathematics and physics, the radian is a unit of angle measure.
	It is the SI derived unit of angle.
	It is defined as the angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle.

More results at FactBites »

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