Radian

An angle of 1 radian results in an arc with a length equal to the radius of the circle.

The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. 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 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.

1 Definition
2 History
3 Conversions
- 3.1 Conversion between radians and degrees
- 3.2 Conversion between radians and grads
4 Advantages of measuring in radians
5 Dimensional analysis
6 Use in physics
7 Multiples of radian units
8 See also
9 References
10 External links

Definition

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 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

History

In centuries before, the concept of radian was used by astronomers to measure the distance between the stars as a function or factor of distance from earth to a particular star (R).In this way a kind of unity was made when they wanted to talk about the distance between different stars. for instance, they said the distance from star s1 and star s2 is 2R(2 times the radius or 2 times the distance between the earth and a particular star).In this way they wouldn't use the numbers or difficult units. Dimensionless numbers also used in many other areas in science and engineering, have many advantages that you can refer to if you wish.

In more modern science the name of radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714.^[1] 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.^[2]^[3]^[4]

Conversions

Conversion between radians and degrees

A chart to convert between degrees and radians

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

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

For example:

$1 \mbox{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ$

$2.5 \mbox{ rad} = 2.5 \cdot \frac {180^\circ} {\pi} \approx 143.2394^\circ$

$\frac {\pi} {3} \mbox{ rad} = \frac {\pi} {3} \cdot \frac {180^\circ} {\pi} = 60^\circ$

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

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

For example:

$1^\circ = 1 \cdot \frac {\pi} {180^\circ} \approx 0.0175 \mbox{ rad}$

$23^\circ = 23 \cdot \frac {\pi} {180^\circ} \approx 0.4014 \mbox{ rad}$

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

Conversion between radians and grads

2π radians are equal to one complete revolution, which is 400^g. So, to convert from radians to grads multiply by 200/π, and to convert from grads to radians multiply by π/200. For example,

$1.2 \mbox{ rad} = 1.2 \cdot \frac {200^{\rm g}} {\pi} \approx 76.3944^{\rm g}$

$50^{\rm g} = 50 \cdot \frac {\pi} {200^{\rm g}} \approx 0.7854 \mbox{ rad}$

The table shows the conversion of some common angles.

Units	Values
Revolutions	0	1/12	1/8	1/6	1/4	1/2	3/4	1
Degrees	0°	30°	45°	60°	90°	180°	270°	360°
Radians	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\pi\,$	$\frac{3\pi}{2}$	$2\pi\,$
Grads	0^g	$\frac{100}{3}^{\rm g}$	50^g	$\frac{200^{\rm g}}{3}$	100^g	200^g	300^g	400^g

Advantages of measuring in radians

Some common angles, measured in radians. All the polygons are regular polygons.

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

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

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

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

$\frac{d^2}{dx^2} \sin x = -\sin x.$

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

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

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 = πx /180, so

$\sin x_\mathrm{deg} = \sin y_\mathrm{rad} = \frac{\pi}{180} x - \left (\frac{\pi}{180} \right )^3\ \frac{x^3}{3!} + \left (\frac{\pi}{180} \right )^5\ \frac{x^5}{5!} - \left (\frac{\pi}{180} \right )^7\ \frac{x^7}{7!} + \cdots .$

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:

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

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

References

↑ O'Connor, J.J. and E.F. Robertson (February 2005). "Biography of Roger Cotes". The MacTutor History of Mathematics. http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html.
↑ Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147–148
↑ Nature, 1910, Vol. 83, pp. 156, 217, and 459–460
↑ Earliest Known Uses of Some of the Words of Mathematics

External links

Radian at MathWorld

SI units

Base units	Ampere · Candela · Kelvin · Kilogram · Metre · Mole · Second

Derived units	Becquerel · Coulomb · degree Celsius · Farad · Gray · Henry · Hertz · Joule · Katal · Lumen · Lux · Newton · Ohm · Pascal · Radian · Siemens · Sievert · Steradian · Tesla · Volt · Watt · Weber

Accepted for use with SI	Atomic mass unit · Astronomical unit · Day · Decibel · Degree of arc · Electronvolt · Hectare · Hour · Litre · Minute · Minute of arc · Neper · Second of arc · Tonne Atomic units · Natural units

See also	SI prefixes · Systems of measurement · Conversion of units

Book:International System of Units · Category:SI base units