Radian

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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".

1 Definition
2 History
3 Explanation
4 Dimensional analysis
5 SI multiples
6 See also
7 External links

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Definition

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.

1 rad = m·m^–1 = 1

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

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History

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; [1]).

The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714 [2]. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.

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Explanation

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.

Angle measures in radians are often given without any explicit unit. When a unit is given, sometimes the symbol rad is used, sometimes the symbol ^c (for "circular"). Care must be taken with this symbol since it can be mistaken for the ° (degree) symbol.

Image:Radian cropped color.png

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

<math>1 \mbox{ rad} = \frac {360^\circ} {2 \pi} = \frac {180^\circ} {\pi} \approx 57.29577951^\circ </math>

or:

<math>1^\circ=\frac{2\pi}{360}\mbox{ rad}=\frac{\pi}{180}\mbox{ rad} \approx 0.01745329\mbox{ rad}</math>

More generally, we can say:

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

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

<math>\lim_{h\rightarrow 0}\frac{\sin h}{h}=1</math>,

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

<math>\frac{d}{dx} \sin x = \cos x</math>.

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.

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Dimensional analysis

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 <math>\sin x</math>:

<math>\sin x = x - \frac{x^3}{3!} + \cdots</math>

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

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SI multiples

SI prefixes have limited use with radians. The milliradian (0.001 rad) is used in gunnery in some countriesTemplate:Citeneeded, because it corresponds to 1 m at a range of 1000 m. Similarly, the prefixes smaller than milli- are potentially usefully in measuring extremely small angles. However, the larger prefixes have no apparent utility.

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