Radian

radiansradmicroradianmradμrad1Concept of the radiannanoradian
[[File:Circle radians.gif|thumb|right|300px|An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian.wikipedia
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Angle

acute angleobtuse angleoblique
The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at ). 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.
The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians.

Degree (angle)

°degreesdegree
The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at ).
It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.

International System of Units

SISI unitsSI unit
The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.
The radian, being 1⁄2π of a revolution, has mathematical advantages but is rarely used for navigation.

SI derived unit

derived unitderived unitsJ/kg
The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.
The International System of Units assigns special names to 22 derived units, which includes two dimensionless derived units, the radian (rad) and the steradian (sr).

Arc (geometry)

arccircular arcarcs
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.
If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than radians.

Arc length

rectifiable curvearclengthlength
The relation can be derived using the formula for arc length.

Trigonometric functions

cosinetrigonometric functiontangent
Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians.
In a right angled triangle, the sum of the two acute angles is a right angle, that is 90° or \frac \pi 2 radians.

Polar coordinate system

polar coordinatespolarpolar coordinate
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.
Angles in polar notation are generally expressed in either degrees or radians (2[[pi|]] rad being equal to 360°).

Euler's formula

complex exponentialcomplex exponentialsEuler's exponential formula
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.
are the trigonometric functions cosine and sine respectively, with the argument x given in radians.

Angular velocity

angular speedangular velocitiesOrders of magnitude (angular velocity)
For example, angular velocity is typically measured in radians per second (rad/s).
If angle is measured in radians, the linear velocity is the radius times the angular velocity, v = r\omega.

Spherical coordinate system

spherical coordinatessphericalspherical polar coordinates
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.
The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system.

Milliradian

milmradangular mil
A milliradian (mrad) is a thousandth of a radian and a microradian (μrad) is a millionth of a radian, i.e. 1 rad = 10 3 mrad = 10 6 μrad.
A milliradian, often called a mil or mrad, is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian).

Radian per second

radians per secondrad/sradians/second
For example, angular velocity is typically measured in radians per second (rad/s).
The radian per second is also the unit of angular frequency.

Phase (waves)

phasephase shiftout of phase
Likewise, the phase difference of two waves can also be measured in radians.
If it is expressed in radians, the same increase in t will increase the phase by 2\pi.

Dimensionless quantity

dimensionlessdimensionless numberdimensionless quantities
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.
Other common proportions are percentages % (= 0.01), ‰ (= 0.001) and angle units such as radians, degrees (°= undefined⁄180) and grads(= undefined⁄200).

Roger Cotes

Coates [''sicCotes, RogerR. Cotes
The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714.

Minute and second of arc

masarcsecondarc second
More common is arc second, which is undefined⁄648,000 rad (around 4.8481 microradians).
A minute of arc is undefined⁄10,800 of a radian.

Angular acceleration

rotational accelerationradian per second squared1901
Similarly, angular acceleration is often measured in radians per second per second (rad/s 2 ).
In SI units, it is measured in radians per second squared (rad/s), and is usually denoted by the Greek letter alpha .

Taylor series

Taylor expansionMaclaurin seriesTaylor polynomial
The trigonometric functions also have simple and elegant series expansions when radians are used; for example, the following Taylor series for sin x :
All angles are expressed in radians.

Turn (angle)

turnturnsrevolution
Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2. 2\pi radians equals one turn, which is by definition 400 gradians (400 gons or 400 g ).
A turn is a unit of plane angle measurement equal to 2degrees or 400 gradians.

Angular frequency

angular rateangular speedangular frequencies
One revolution is equal to 2π radians, hence

Steradian

srmillisteradiansmsr
It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles.

Metric prefix

SI prefixunit prefixprefix
Metric prefixes have limited use with radians, and none in mathematics.
The SI unit of angle is the radian, but degrees, minutes, seconds and more rarely tierces see some scientific use.

Gradian

gradgongrads
2\pi radians equals one turn, which is by definition 400 gradians (400 gons or 400 g ).
It is equivalent to 1⁄400 of a turn, 9⁄10 of a degree, or undefined⁄200 of a radian.

Pi

ππ\pi
There are 2[[pi|]] × 1000 milliradians (≈ 6283.185 mrad) in a circle.
plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2 radians.