Radian

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See Radian (band) for the Austrian trio.


The radian (symbol: rad) is the SI unit of plane angle.

Definition

1 rad = m·m–1 = 1

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.

The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle is one radian. Angle measures in radians are often given without any explicit unit. When a unit is given, sometimes the abbreviation rad is used, sometimes the symbol c (for "circular"). Care must be taken with this symbol since it can be mistaken for the ° (degree) symbol.

File:Radian cropped color.png
An angle measuring 1 radian subtends an arc equal in length to the radius of the circle.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi\mbox{rad} = 360^\circ}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \mbox{rad} = \frac {360^\circ} {2 \pi} = \frac {180^\circ} {\pi} \approx 57.29577951^\circ }

or:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 360^\circ=2\pi\mbox{rad}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1^\circ=\frac{2\pi}{360}\mbox{rad}=\frac{\pi}{180}\mbox{rad} \approx 0.01745329\mbox{rad}}

More generally, we can say:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \mbox{rad} = x \frac {180^\circ} {\pi}}

If, for example, -1,570796 in radians was given, the corresponding degree value would be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1.570796 \mbox{rad} = -1.570796 \cdot \frac {180^\circ} {\pi} = -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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{h\rightarrow 0}\frac{\sin h}{h}=1} ,

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x = \cos x} .

The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995.

For measuring solid angles, see steradian.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x) = x - \frac{x^3}{3!} + \cdots}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} had units, then the sum would be meaningless; the linear term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} can not be added to the cubic term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3/3!} , etc. Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} must be dimensionless.

See also

bg:Радиан ca:Radiant (angle) cs:Radián da:Radian de:Bogenmaß et:Radiaan es:Radián fr:Radian gl:Radián ko:라디안 it:Radiante he:רדיאן nl:Radiaal ja:ラジアン no:Radian pl:Radian pt:Radiano ru:Радиан sl:Radian sr:Радијан fi:Radiaani sv:Radian uk:Радіан