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## Conversions

(File:Degree-Radian Conversion.svg|thumb|300px|A chart to convert between degrees and radians){{Table of angles}}

### Conversion between radians and degrees

As stated, one radian is equal to 180/{{pi}} degrees. Thus, to convert from radians to degrees, multiply by 180/{{pi}}.
text{angle in degrees} = text{angle in radians} cdot frac {180^circ} {pi}
For example:
1 text{ rad} = 1 cdot frac {180^circ} {pi} approx 57.2958^circ 2.5 text{ rad} = 2.5 cdot frac {180^circ} {pi} approx 143.2394^circ frac {pi} {3} text{ rad} = frac {pi} {3} cdot frac {180^circ} {pi} = 60^circ
Conversely, to convert from degrees to radians, multiply by {{pi}}/180.
text{angle in radians} = text{angle in degrees} cdot frac {pi} {180^circ}
For example:
1^circ = 1 cdot frac {pi} {180^circ} approx 0.0175 text{ rad}
23^circ = 23 cdot frac {pi} {180^circ} approx 0.4014 text{ rad}Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2{{pi}}.

#### Radian to degree conversion derivation

The length of circumference of a circle is given by 2pi r, where r is the radius of the circle.So the following equivalent relation is true:360^circ iff 2pi r{{pad|4em}}[Since a 360^circ sweep is needed to draw a full circle]By the definition of radian, a full circle represents:
frac{2pi r}{r} text{ rad}
= 2pi text{ rad}
Combining both the above relations:
2pi text{ rad} = 360^circ
Rrightarrow 1 text{ rad} = frac{360^circ}{2pi}
Rrightarrow 1 text{ rad} = frac{180^circ}{pi}

2pi radians equals one turn, which is by definition 400 gradians (400 gons or 400g). So, to convert from radians to gradians multiply by 200/pi, and to convert from gradians to radians multiply by pi/200. For example,
1.2 text{ rad} = 1.2 cdot frac {200^text{g}} {pi} approx 76.3944^text{g} 50^text{g} = 50 cdot frac {pi} {200^text{g}} approx 0.7854 text{ rad}

File:Radian-common.svg|thumb|357px|right|Some common angles, measured in radians. All the large polygons in this diagram are regular polygonregular polygonIn 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_{hrightarrow 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 {{pi}}/180: if x is the number of degrees, the number of radians is {{nowrap|1=y = {{pi}}x / 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.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.For a debate on this meaning and use see:JOURNAL, 10.1119/1.18616, Anglesâ€”Let's treat them squarely, 1997, Brownstein, K. R., American Journal of Physics, 65, 7, 605, 1997AmJPh..65..605B, ,JOURNAL, Angles as a fourth fundamental quantity, 1962, Romain, J.E., Journal of Research of the National Bureau of Standards Section B, 66B, 3, 97,weblink ,JOURNAL, 10.1119/1.18964, Dimensional angles and universal constants, 1998, LÃ©Vy-Leblond, Jean-Marc, American Journal of Physics, 66, 9, 814, 1998AmJPh..66..814L, , and JOURNAL, 10.1119/1.19185, Unitsâ€”SI-Only, or Multicultural Diversity?, 1999, Romer, Robert H., American Journal of Physics, 67, 13, 1999AmJPh..67...13R,

## 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 2{{pi}} 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 sâˆ’1 and sâˆ’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 (kâ‹…2{{pi}}) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is ({{nowrap|kâ‹…2{{pi}} + {{pi}}}}), where k is an integer, they are considered in antiphase.

## Notes and references

{{Wiktionary|radian}} {{SI units}}{{UCUM units}}

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- time: 1:20pm EDT - Tue, Mar 19 2019
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