Radian

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

Some common angles, measured in radians.

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 ".

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995 and the radian is now considered an SI derived unit. For measuring solid angles, see steradian.

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.

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

[edit] Definition

An angle measuring 1 radian subtends an arc equal in length to the radius of the circle.

One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle.

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.

[edit] 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 the University of St Andrews, 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.

[edit] 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.

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

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

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

so:

$0.01745329\mbox{ rad} \approx 1.0^\circ$

More generally, we can say:

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

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

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

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

$\lim_{h\rightarrow 0}\frac{\sin h}{h}=1$ ,

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

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

[edit] 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 sin x:

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

If x had units, then the sum would be meaningless; the linear term x cannot be added to (or have subtracted) the cubic term $x 3 / 3!$ . Therefore, x must be dimensionless.

[edit] SI multiples

SI prefixes have limited use with radians. The milliradian (0.001 rad, or 1 mrad) is used in gunnery and general targeting, because it corresponds to 1 m at a range of 1000 m (at such small angles, the curvature can be considered 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. However, the larger prefixes have no apparent utility, mainly because to exceed 2π radians is to begin the same circle (or revolutionary cycle) again.

[edit] External links

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

Radian

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] History

[edit] Explanation

[edit] Dimensional analysis

[edit] SI multiples

[edit] External links

[edit] See also

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