Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit. The SI unit of solid angle measurement is the steradian.

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°"). As the ratio of two lengths, the radian is 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.

Definition

Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. 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 such a subtended angle is equal to the ratio of the arc length 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 2π''r'' /''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 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.

Conversions

Conversion between radians and degrees

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

:$\backslash mbox\{deg\}\; =\; \backslash mbox\{rad\}\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}$

For example: :$1\; \backslash mbox\{\; rad\}\; =\; 1\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; 57.2958^\backslash circ$ :$2.5\; \backslash mbox\{\; rad\}\; =\; 2.5\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; 143.2394^\backslash circ$ :$\backslash frac\; \{\backslash pi\}\; \{3\}\; \backslash mbox\{\; rad\}\; =\; \backslash frac\; \{\backslash pi\}\; \{3\}\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; =\; 60^\backslash circ$

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

:$\backslash mbox\{rad\}\; =\; \backslash mbox\{deg\}\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180^\backslash circ\}$

For example:

:$1^\backslash circ\; =\; 1\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180^\backslash circ\}\; \backslash approx\; 0.0175\; \backslash mbox\{\; rad\}$ $23^\backslash circ\; =\; 23\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{180^\backslash circ\}\; \backslash approx\; 0.4014\; \backslash mbox\{\; rad\}$

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

Radian to degree conversion derivation

We know that the length of circumference of a circle is given by $2\backslash pi\; r$, where $r$ is the radius of the circle.

So, we can very well say that the following equivalent relation is true:

$360^\backslash circ\; \backslash iff\; 2\backslash pi\; r$[Since a $360^\backslash circ$ sweep is need to draw a full circle]

By definition of radian, we can formulate that a full circle represents:

:$\backslash frac\{2\backslash pi\; r\}\{r\}\; \backslash mbox\{\; rad\}$

:$=\; 2\backslash pi\; \backslash mbox\{\; rad\}$

Combining both the above relations we can say:

:$2\backslash pi\; \backslash mbox\{\; rad\}\; =\; 360^\backslash circ$

:$\backslash Rrightarrow\; 1\; \backslash mbox\{\; rad\}\; =\; \backslash frac\{360^\backslash circ\}\{2\backslash pi\}$

:$\backslash Rrightarrow\; 1\; \backslash mbox\{\; rad\}\; =\; \backslash frac\{180^\backslash circ\}\{\backslash pi\}$

Conversion between radians and grads

$2\backslash pi$ radians are equal to one turn, which is 400^g. So, to convert from radians to grads multiply by $200/\backslash pi$, and to convert from grads to radians multiply by $\backslash pi/200$. For example,

:$1.2\; \backslash mbox\{\; rad\}\; =\; 1.2\; \backslash cdot\; \backslash frac\; \{200^\{\backslash rm\; g\}\}\; \{\backslash pi\}\; \backslash approx\; 76.3944^\{\backslash rm\; g\}$ :$50^\{\backslash rm\; g\}\; =\; 50\; \backslash cdot\; \backslash frac\; \{\backslash pi\}\; \{200^\{\backslash rm\; g\}\}\; \backslash approx\; 0.7854\; \backslash mbox\{\; rad\}$

The table shows the conversion of some common angles.

{|class = wikitable ! Units !! colspan=8 | Values |- valign="top" |style="background:#f2f2f2" | Turns   |style="width:3em; text-align:center" | 0 |style="width:3em; text-align:center" | $\backslash tfrac\{\backslash tau\}\{12\}$ |style="width:3em; text-align:center" | $\backslash tfrac\{\backslash tau\}\{8\}$ |style="width:3em; text-align:center" | $\backslash tfrac\{\backslash tau\}\{6\}$ |style="width:3em; text-align:center" | $\backslash tfrac\{\backslash tau\}\{4\}$ |style="width:3em; text-align:center" | $\backslash tfrac\{\backslash tau\}\{2\}$ |style="width:3em; text-align:center" | $\backslash tfrac\{3\backslash tau\}\{4\}$ |style="width:3em; text-align:center" | $\backslash tau$ |- valign="top" |style="background:#f2f2f2" | Degrees   |style="width:3em; text-align:center" | 0° |style="width:3em; text-align:center" | 30° |style="width:3em; text-align:center" | 45° |style="width:3em; text-align:center" | 60° |style="width:3em; text-align:center" | 90° |style="width:3em; text-align:center" | 180° |style="width:3em; text-align:center" | 270° |style="width:3em; text-align:center" | 360° |- valign="top" |style="background:#f2f2f2" | Radians |style="text-align:center" | 0 |style="text-align:center" | $\backslash tfrac\{\backslash pi\}\{6\}$ |style="text-align:center" | $\backslash tfrac\{\backslash pi\}\{4\}$ |style="text-align:center" | $\backslash tfrac\{\backslash pi\}\{3\}$ |style="text-align:center" | $\backslash tfrac\{\backslash pi\}\{2\}$ |style="text-align:center" | $\backslash pi$ |style="text-align:center" | $\backslash tfrac\{3\backslash pi\}\{2\}$ |style="text-align:center" | 2$\backslash pi$ |- valign="top" |style="background:#f2f2f2" | Grads |style="text-align:center" | 0^g |style="text-align:center" | $\backslash tfrac\{100^g\}\{3\}$ |style="text-align:center" | 50^g |style="text-align:center" | $\backslash tfrac\{200^g\}\{3\}$ |style="text-align:center" | 100^g |style="text-align:center" | 200^g |style="text-align:center" | 300^g |style="text-align:center" | 400^g |}

Since a turn $\backslash tau$ is often identified with the circle constant $2\backslash pi$ there is essentially no difference between measuring angles in turns and in radians.

Advantages of measuring in radians

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

:$\backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{\backslash sin\; h\}\{h\}=1,$

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

:$\backslash frac\{d\}\{dx\}\; \backslash sin\; x\; =\; \backslash cos\; x$ :$\backslash frac\{d^2\}\{dx^2\}\; \backslash sin\; x\; =\; -\backslash sin\; x.$

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 $\backslash frac\{d^2\; y\}\{dx^2\}\; =\; -y$, the evaluation of the integral $\backslash int\; \backslash frac\{dx\}\{1+x^2\}$, 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'' :

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

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'' = π''x'' /180, so

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

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:

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

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 (k·2π) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is (k·2π + π), where k is an integer, 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 $\backslash pi$ = 3.2, there are 6400 mils in a complete rotation. Other gunnery systems may use a different approximation to $\backslash pi$. 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.

Criticism

The widespread habit of expressing radians in terms of fractions of $\backslash pi$ has been criticized because in this way one essentially use $\backslash pi$ as a unit. An angle of $\backslash pi$ radians represents half a circle, which appear to be a rather arbitrary choice of unit. Therefore various mathematicians and scientists have been critical of the custom of expressing radians as fractions of , and have proposed to use other constants instead. A. Eagle proposed to have a constant representing one quarter-turn. A constant representing 2 has been proposed because a circle is 2 radians around. Robert Palais proposed to use a "pi with three legs" ($\backslash pi\backslash !\backslash !\backslash !\backslash !\backslash ;\backslash pi$) to denote 1 turn, while several people independently have proposed to use the Greek letter (tau) instead.

See also

Angular mil - military measurement

Trigonometry

Harmonic analysis

Angular frequency

Grad

Steradian - the "square radian"

References

External links

Radian at MathWorld

Category:Natural units Category:SI derived units Category:Trigonometry Category:Units of angle

af:Radiaal ar:راديان be-x-old:Радыян bo:གཞུ་ཚད། bs:Radijan bg:Радиан ca:Radian cs:Radián cy:Radian da:Radian de:Radiant (Einheit) et:Radiaan el:Ακτίνιο (μονάδα μέτρησης) es:Radián eo:Radiano eu:Radian fa:رادیان fr:Radian gl:Radián gan:弧度 ko:라디안 hr:Radijan id:Radian it:Radiante he:רדיאן ka:რადიანი kk:Радиан lv:Radiāns lt:Radianas mk:Радијан mr:त्रिज्यी ms:Radian nl:Radiaal (wiskunde) ja:ラジアン no:Radian nn:Radian pl:Radian pt:Radiano ro:Radian ru:Радиан si:රේඩියනය simple:Radian sk:Radián sl:Radian sr:Радијан sh:Radijan fi:Radiaani sv:Radian ta:ஆரையம் th:เรเดียน tg:Радиан uk:Радіан zh:弧度

Coordinates	52°05′36″N5°7′10″N
name	Sean Price
background	solo_singer
alias
birth date	March 17, 1972
origin	Brownsville, Brooklyn, New York
genre	Hip Hop
years active	1993-present
label	Duck Down
associated acts	Boot Camp Clik
website	}}

Sean Price (born March 17, 1972) is an American rapper and member of the hip hop supergroups Boot Camp Clik and Random Axe. He came to fame as one-half of the duo Heltah Skeltah, performing under the name Ruckus, or in short Ruck, along with partner Rock. He also is featured in the video game NBA 2K11 as a playable character in street mode. Price lives in Brownsville, Brooklyn.

Discography

Solo albums

Title	Album details	Peak chart positions
		! scope="col" style="width:2.2em;font-size:90%;"	! scope="col" style="width:2.2em;font-size:90%;"	! scope="col" style="width:2.2em;font-size:90%;"
		*Released: June 13, 2005	*Label: Duck Down (#2011)	*Format: CD, digital download, LP
		*Released: January 30, 2007	*Label: Duck Down (#2045)	*Format: CD, digital download
		*Released: TBA 2011	*Label: Duck Down
		Singles

+ List of singles, with selected chart positions	Title	Year	Peak chart positions	Album
! scope="col" style="width:3em;font-size:90%;"

Mixtapes

''Donkey Sean Jr.'' (2004)

''Master P'' (2007)

''Kimbo Price'' (2009)

''The Price Is Right'' (2010)

''The Price Is Right Vol. 2'' (2011)

References

External links

Duck Down Records website

Category:1972 births Category:Living people Category:African American rappers Category:Rappers from New York City Category:People from Brooklyn Category:Underground rappers Category:Boot Camp Clik members

es:Sean Price (rapero) fr:Sean Price it:Sean Price hu:Sean Price fi:Sean Price uk:Sean Price