Radian

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. 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.

While it is normally asserted that as the ratio of two lengths, the radian is a "pure number", Mohr and Phillips[2] dispute this assertion. However, in mathematical writing the symbol "rad" is almost always omitted[2]. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used. The radian is defined as 1.[3] There is controversy as to whether it is satisfactory in the SI to consider angles to be dimensionless since they can be measured in degrees, radians or cycles.[4] This can lead to problems when considering the units for frequency and the Planck constant.[2][5][6][7]

A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π).

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.

The relation ${\displaystyle 2\pi {\text{ rad}}=360^{\circ }}$ can be derived using the formula for arc length. Taking the formula for arc length, or ${\displaystyle \ell _{arc}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)}$. Assuming a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that ${\displaystyle 1=2\pi \left({\frac {1{\text{ rad}}}{360^{\circ }}}\right)}$. This can be further simplified to ${\displaystyle 1={\frac {2\pi {\text{ rad}}}{360^{\circ }}}}$. Multiplying both sides by ${\displaystyle 360^{\circ }}$ gives ${\displaystyle 360^{\circ }=2\pi {\text{ rad}}}$.

The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714.[8][9] He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure. Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.[10]

The idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi (c. 1400) used so-called diameter parts as units where one diameter part was 1/60 radian and they also used sexagesimal subunits of the diameter part.[11]

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 had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between the terms rad, radial, and radian. In 1874, after a consultation with James Thomson, Muir adopted radian.[12][13][14] The name radian was not universally adopted for some time after this. Longmans' School Trigonometry still called the radian circular measure when published in 1890.[15]

The International Bureau of Weights and Measures[16] and International Organization for Standardization[17] specify rad as the symbol for the radian. Alternative symbols used 100 years ago are ^c (the superscript letter c, for "circular measure"), the letter r, or a superscript ^R,[18] but these variants are infrequently used as they may be mistaken for a degree symbol (°) or a radius (r). So, for example, a value of 1.2 radians would most commonly be written as 1.2 rad; other notations include 1.2 r, 1.2^rad, 1.2^c, or 1.2^R.

Some common angles, measured in radians. All the large polygons in this diagram 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

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

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

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

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

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

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

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

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

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.

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.

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.[19]

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 of angular velocity and angular acceleration 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 (k⋅2π) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is (k⋅2π + π), where k is an integer, they are considered in antiphase.

Metric prefixes have limited use with radians, and none in mathematics. A milliradian (mrad) is a thousandth of a radian and a microradian (μrad) is a millionth of a radian, i.e. 1 rad = 10³ mrad = 10⁶ μrad.

There are 2π × 1000 milliradians (≈ 6283.185 mrad) in a circle. So a milliradian is just under 1/6283 of the angle subtended by a full circle. This "real" unit of angular measurement of a circle is in use by telescopic sight manufacturers using (stadiametric) rangefinding in reticles. The divergence of laser beams is also usually measured in milliradians.

An approximation of the milliradian (0.001 rad) is used by NATO and other military organizations in gunnery and targeting. Each angular mil represents 1/6400 of a circle and is 15/8% or 1.875% smaller than the milliradian. For the small angles typically found in targeting work, the convenience of using the number 6400 in calculation outweighs the small mathematical errors it introduces. In the past, other gunnery systems have used different approximations to 1/2000π; for example Sweden used the 1/6300 streck and the USSR used 1/6000. Being based on the milliradian, the NATO mil subtends roughly 1 m at a range of 1000 m (at such small angles, the curvature is negligible).

Smaller units like microradians (μrad) and nanoradians (nrad) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is arc second, which is π/648,000 rad (around 4.8481 microradians). Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles.

• Angular frequency
• Steradian, a higher-dimensional analog of the radian which measures solid angle
• Trigonometry

Wikibooks has a book on the topic of: Trigonometry/Radian and degree measures

Look up radian in Wiktionary, the free dictionary.

• Media related to Radian at Wikimedia Commons
• Radian at MathWorld