Editing Radian (section)

== Usage ==

=== Mathematics ===
[[File:Radian-common.svg|thumb|357px|right|Some common angles, measured in radians. All the large polygons in this diagram are [[regular polygon]]s.]]

In [[calculus]] and most other branches of mathematics beyond practical [[geometry]], angles are measured in radians. This is because radians have a mathematical naturalness that leads to a more elegant formulation of some important results.

Results in [[analysis (mathematics)|analysis]] involving [[trigonometric function]]s can be elegantly stated when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple [[limit of a function|limit]] formula

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

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

:<math>\frac{d}{dx} \sin x = \cos x</math>
:<math>\frac{d^2}{dx^2} \sin x = -\sin x.</math>

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]] <math> \tfrac{d^2 y}{dx^2} = -y </math>, the evaluation of the integral <math> \textstyle\int \frac{dx}{1+x^2}, </math> and so on). In all such cases, it is appropriate that the arguments of the functions are treated as (dimensionless) numbers—without any reference to angles.

The trigonometric functions of angles also have simple and elegant series expansions when radians are used. For example, when ''x'' is the angle expressed in radians, the [[Taylor series]] for sin&nbsp;''x'' becomes:

:<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots .</math>

If ''y'' were the angle ''x'' but expressed in degrees, i.e. {{nowrap|1=''y'' = {{pi}}''x'' / 180}}, then the series would contain messy factors involving powers of {{pi}}/180:

:<math>\sin y = \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 .</math>

In a similar spirit, if angles are involved, mathematically important relationships between the sine and cosine functions and the [[exponential function]] (see, for example, [[Euler's formula]]) can be elegantly stated when the functions' arguments are angles expressed in radians (and messy otherwise). More generally, in complex-number theory, the arguments of these functions are (dimensionless, possibly complex) numbers—without any reference to physical angles at all.

=== Physics ===
The radian is widely used in [[physics]] when angular measurements are required. For example, [[angular velocity]] is typically expressed in the unit [[radian per second]] (rad/s). One revolution per second corresponds to 2{{pi}} radians per second.

Similarly, the unit used for [[angular acceleration]] is often radian per second per second (rad/s<sup>2</sup>).

For the purpose of [[dimensional analysis]], the units of angular velocity and angular acceleration are s<sup>−1</sup> and s<sup>−2</sup> respectively.

Likewise, the phase angle difference of two waves can also be expressed using the radian as the unit. For example, if the phase angle difference of two waves is (''n''⋅2{{pi}}) radians, where ''n'' is an integer, they are considered to be in [[phase (waves)|phase]], whilst if the phase angle difference of two waves is ({{nowrap|''n''⋅2{{pi}} + {{pi}}}}) radians, with ''n'' an integer, they are considered to be in antiphase.

A unit of reciprocal radian or inverse radian (rad<sup>−1</sup>) is involved in derived units such as meter per radian (for [[angular wavelength]]) or newton-metre per radian (for torsional stiffness).

=== Prefixes and variants ===
[[Metric prefix]]es for submultiples are used with radians. A [[milliradian]] (mrad) is a thousandth of a radian (0.001&nbsp;rad), i.e. {{nowrap|1=1 rad = 10<sup>3</sup> mrad}}. There are 2[[pi|{{pi}}]] × 1000 milliradians (≈ 6283.185&nbsp;mrad) in a circle. So a milliradian is just under {{sfrac|1|6283}} of the angle subtended by a full circle. This unit of angular measurement of a circle is in common use by [[telescopic sight]] manufacturers using [[Stadiametric rangefinding|(stadiametric) rangefinding]] in [[reticle]]s. The [[beam divergence|divergence]] of [[laser]] beams is also usually measured in milliradians.

The [[angular mil]] is an approximation of the milliradian used by [[NATO]] and other military organizations in [[gun]]nery and [[Sniper#Targeting, tactics and techniques|targeting]]. Each angular mil represents {{sfrac|1|6400}} of a circle and is {{sfrac|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 {{sfrac|1|2000{{pi}}}}; for example Sweden used the {{sfrac|1|6300}} ''streck'' and the USSR used {{sfrac|1|6000}}. Being based on the milliradian, the NATO mil subtends roughly 1&nbsp;m at a range of 1000&nbsp;m (at such small angles, the curvature is negligible).

Prefixes smaller than milli- are useful in measuring extremely small angles. Microradians (μrad, {{val|e=-6|u=rad}}) and nanoradians (nrad, {{val|e=-9|u=rad}}) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. More common is the [[arc second]], which is {{sfrac|{{pi}}|648,000}}&nbsp;rad (around 4.8481&nbsp;microradians).

{{SI multiples
| unit=radian
| symbol=rad
| xm=[[milliradian]]
}}