Radian

For the musical group, see Radian (band). See Radian for the mathematical concept or Radian (comics for the comic book character For the comic book character, see Radian (comics). See Radian for the mathematical concept or Radian (band for the musical group Christian Cord codenamed Radian and later Phaser

Some common angles, measured in radians. All the polygons are regular polygons.

The radian is a unit of plane angle, equal to 180/π degrees, or about 57. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems This article describes the unit of angle For other meanings see Degree. 2958 degrees. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

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 (° Unicode: U+00B0 HTML: &deg is a typographical symbol or Glyph, that is used to represent degrees of arc (see

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. Until 1995, SI ( International System of Units) supplementary units were QuantitySymbolName of SI supplementary SI derived units are part of the SI system of measurement units and are derived from the seven SI base units They are derived from SI basic units/defined The SI unit of solid angle measurement is the steradian. The solid angle, Ω, is the angle in three-dimensional space that an object Subtends at a point The steradian (symbol sr) is the SI unit of Solid angle. It is used to describe two-dimensional angular spans in three- Dimensional space

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. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

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. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Geometry, an arc subtended by an Angle is a Curve whose endpoints are on the angle's two rays Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Geometry, an arc is a closed segment of a Differentiable Curve in the two-dimensional plane; for example a circular Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

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. Roger Cotes FRS ( July 10, 1682 – June 5, 1716) was an English Mathematician, known for working closely with ^[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. Events 70 - Titus and his Roman Legions breach the middle wall of Jerusalem in the Siege of Jerusalem Year 1873 ( MDCCCLXXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common James Thomson ( February 16, 1822 - May 8, 1892) was an Irish Engineer and Physicist whose reputation would have William Thomson 1st Baron Kelvin (or Lord Kelvin) OM, GCVO, PC, PRS, FRSE, (26 June 1824 &ndash 17 December 1907 Queen's University Belfast is a university in Belfast, Northern Ireland. Belfast ( is the capital city of Northern Ireland and the seat of government in Northern Ireland. 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. Thomas Muir may refer to Thomas Muir (radical (1765-1799 leader of the Scottish Friends of the People Society The University of St Andrews is the oldest University in Scotland and third oldest in the English-speaking world, having been founded between In 1874, Muir adopted radian after a consultation with James Thomson. Year 1874 ( MDCCCLXXIV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Common ^[2]^[3]^[4]

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

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. The grad is a unit of plane Angle, equivalent to of a full Circle, dividing a Right angle in 100 For example,

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

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

Reasons why radians are preferred in mathematics

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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. Analysis has its beginnings in the rigorous formulation of Calculus. 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 d²y/dx² = −y, the evaluation of the integral ∫dx/(1 + x²), and so on). In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" 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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

Dimensional analysis

Although the radian is a unit of measure, it is a dimensionless quantity. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units 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:

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

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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives 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). Do not confuse with Angular frequency The unit for angular velocity is 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). Angular acceleration is the rate of change of Angular velocity over Time.

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

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). READ DISCUSSION PAGE BEFORE MAKING ANY EDITS TO CAPTION BELOW http//en The divergence of laser beams is also usually measured in milliradians. The beam divergence of an electromagnetic beam is an angular measure of the increase in Beam diameter with distance from the optical aperture or Antenna aperture A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission.

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. An angular mil, also mil, is a unit of Angle. Origin of the name All versions of the angular mil are approximately the same size as a Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The grad is a unit of plane Angle, equivalent to of a full Circle, dividing a Right angle in 100 This article describes the unit of angle For other meanings see Degree. The steradian (symbol sr) is the SI unit of Solid angle. It is used to describe two-dimensional angular spans in three- Dimensional space 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

MathWorld is an online Mathematics reference work created and largely written by Eric W

Dictionary

-noun

(geometry) 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

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents