Editing Radian (section)

=== 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.