This article is about angles. For the Austrian trio, please see: Radian (band).

The radian is a unit of plane angle. 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 "1.2 rad" or "1.2^c ".

Nowadays, radian is the de facto unit of plane angles for mathematicians, and the symbol "rad" is usually omitted in mathematicial writings. When using degrees, the ° symbol is used to distinguish it from radians.

Definition

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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.

In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle.

History

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The term radian first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between rad, radial and radian. In 1874, Muir adopted radian after a consultation with James Thomson. (Sources: Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147-148; Nature, 1910, Vol. 83, pp. 156, 217, and 459-460; ^*).

The concept of a 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.

Explanation

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The radian is useful to distinguish between quantities of different nature but the same dimension. For example, angular velocity can be measured in radians per second (rad/s). Retaining the word radian emphasizes that angular velocity is equal to 2π times the rotational frequency.

In practice, the symbol rad is used where appropriate, but the derived unit "1" is generally omitted in combination with a numerical value.

There are 2π (approximately 6.28318531) radians in a complete circle, so:

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

$1\; \backslash mbox\{\; rad\}\; =\; \backslash frac\; \{360^\backslash circ\}\; \{2\; \backslash pi\}\; =\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; 57.29577951^\backslash circ$

or:

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

$1^\backslash circ=\backslash frac\{2\backslash pi\}\{360\}\backslash mbox\{\; rad\}=\backslash frac\{\backslash pi\}\{180\}\backslash mbox\{\; rad\}\; \backslash approx\; 0.01745329\backslash mbox\{\; rad\}$

More generally, we can say:

$x\; \backslash mbox\{\; rad\}\; =\; x\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}$

If, for example, -1.570796 in radians was given, the corresponding degree value would be:

$-1.570796\; \backslash mbox\{\; rad\}\; =\; -1.570796\; \backslash cdot\; \backslash frac\; \{180^\backslash circ\}\; \{\backslash pi\}\; \backslash approx\; -90^\backslash circ$

In calculus, angles must be represented in radians in trigonometric functions, to make identities and results as simple and natural as possible. For example, the use of radians leads to the simple identity

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

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

$\backslash frac\{d\}\{dx\}\; \backslash sin\; x\; =\; \backslash cos\; x$.

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995.

For measuring solid angles, see steradian.

Dimensional analysis

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Although the radian is a unit of measure, anything measured in radians is dimensionless. This can be seen easily in that the ratio of an arc's length to its radius is the angle of the arc, measured in radians; yet the quotient of two distances is dimensionless.

Another way to see the dimensionlessness of the radian is in the Taylor series for the trigonometric function sin x:

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

If x had units, then the sum would be meaningless; the linear term x cannot be added to the cubic term $x^3/3!$, etc. Therefore, x must be dimensionless.

SI multiples

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SI prefixes have limited use with radians. The milliradian (0.001 rad) is used in gunnery and general targeting, because it corresponds to 1 m at a range of 1000 m. 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 2 pi radians is to begin the same circle (or revolutionary cycle) again.

See also

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• Trigonometry
• Harmonic analysis
• Angular frequency
• Degree

External links

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• Radian/Degree converter.

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

Радиан | Radiant (angle) | Radián | Radian | Bogenmaß | Radiaan | Radián | Radiano | Radian | Radián | 라디안 | Radiante | רדיאן | Radiaal | ラジアン | Radian | Radian | Radian | Radiano | Радиан | Radián | Radian | Радијан | Radiaani | Radian | Радіан | 弧度