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Radian

From Wikipedia, the free encyclopedia

For other uses, seeRadian (disambiguation).

"mrad" redirects here. For millirads, seeRad (unit).

An angle of 1 radian results in an arc with a length equal to theradiusof thecircle.

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit ofangular measure, used in many areas ofmathematics. The unit was formerly aSI supplementaryunit, but this category was abolished in 1995 and the radian is now considered aSI derived unit.

The SI unit ofsolid anglemeasurement is thesteradian.

The radian is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular

measure"). For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2 c" (the

second symbol is often mistaken for a degree: "1.2"). As the ratio of two lengths, the radian is a

"pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" isalmost always omitted. In the absence of any symbol radians are assumed, and when degrees are

meant the symbolis used.

Contents

[hide]

1 Definition 2 History 3 Conversions

o 3.1 Conversion between radians and degrees 3.1.1 Radian to degree conversion derivation

o 3.2 Conversion between radians and grads 4 Advantages of measuring in radians 5 Dimensional analysis 6 Use in physics

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7 Multiples of radian units 8 Criticism 9 See also 10 References 11 External links

Definition

Radian describes the planeanglesubtendedby a circulararcas the length of the arc divided bytheradiusof the arc. One radian is theanglesubtendedat the center of acircleby anarcthat is

equal in length to theradiusof the circle. More generally, themagnitudein radians of such a

subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, = s /r,

where is the subtended angle in radians, s is arc length, and ris radius. Conversely, the lengthof the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that

is, s = r.

A complete revolution is 2 radians (shown here with a circle of radius one andcircumference

2).

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 2r

/r, or 2. Thus 2 radians is equal to 360

degrees, meaning that one radian is equal to 180/ degrees.

History

The concept of radian measure, as opposed to the degree of an angle, should probably be credited

toRoger Cotesin 1714.[1]

He had the radian in everything but name, and he recognized its

naturalness as a unit of angular measure.

The term radian first appeared in print on 5 June 1873, in examination questions set byJames

Thomson(brother ofLord Kelvin) atQueen's College,Belfast. He used the term as early as

1871, while in 1869,Thomas Muir, then of theUniversity of St Andrews, vacillated betweenrad, radial and radian. In 1874, Muir adopted radian after a consultation with James

Thomson.[2][3][4]

Conversions

Conversion between radians and degrees

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A chart to convert between degrees and radians

As stated, one radian is equal to 180/ degrees. Thus, to convert from radians to degrees,multiply by 180/.

For example:

Conversely, to convert from degrees to radians, multiply by /180.

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For example:

Radians can be converted to turns by dividing the number of radians by 2.

Radian to degree conversion derivation

We know that the length of circumference of a circle is given by 2r, where ris the radius of the

circle.

So, we can very well say that the following equivalent relation is true:

[Since a sweep is need to draw a full circle]

By definition of radian, we can formulate that a full circle represents:

= 2 rad

Combining both the above relations we can say:

Conversion between radians and grads

2 radians are equal to oneturn, which is 400g. So, to convert from radians togradsmultiply by

200 / , and to convert from grads to radians multiply by / 200. For example,

The table shows the conversion of some common angles.

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Units Values

Turns 0

Degrees 0 30 45 60 90 180 270 360

Radians 0 2

Grads 0g 50g 100g 200g 300g 400g

Since aturn is often identified with the circle constant 2 there is essentially no differencebetween measuring angles in turns and in radians.

Advantages of measuring in radians

Some common angles, measured in radians. All the polygons are regular polygons.

Incalculusand most other branches of mathematics beyond practical geometry, angles areuniversally 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 inanalysisinvolvingtrigonometric functionsare simple and elegant when

the functions' arguments are expressed in radians. For example, the use of radians leads to the

simplelimitformula

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

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Because of these and other properties, the trigonometric functions appear in solutions tomathematical problems that are not obviously related to the functions' geometrical meanings (for

example, the solutions to the differential equation , the evaluation of the integral

, and so on). In all such cases it is found that the arguments to the functions are mostnaturally written in the form that corresponds, in geometrical contexts, to the radianmeasurement of angles.

The trigonometric functions also have simple and elegant series expansions when radians areused; for example, the followingTaylor seriesfor sin x :

Ifx were expressed in degrees then the series would contain messy factors involving powers of

/180: ifx is the number of degrees, the number of radians is y= x /180, so

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 adimensionlessquantity. This can be seen from the

definition given earlier: the angle subtended at the centre of a circle, measured in radians, isequal 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.

Another way to see the dimensionlessness of the radian is in the series representations of the

trigonometric functions, such as theTaylor seriesfor sin x mentioned earlier:

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Ifx had units, then the sum would be meaningless: the linear term x cannot be added to (or have

subtracted) the cubic term x3

/ 3! or the quintic term x5

/ 5!, etc. Therefore, x must bedimensionless.

Althoughpolarandspherical coordinatesuse radians to describe coordinates in two and three

dimensions, the unit is derived from the radius coordinate, so the angle measure is stilldimensionless.

Use in physics

The radian is widely used inphysicswhen angular measurements are required. For example,angular velocityis typically measured in radians per second (rad/s). One revolution per second is

equal to 2 radians per second.

Similarly,angular accelerationis often measured in radians per second per second (rad/s2).

For the purpose of dimensional analysis, the units are s1 and s2 respectively.

Likewise, thephase differenceof two waves can also be measured in radians. For example, if the

phase difference of two waves is (k2) radians, where k is an integer, they are considered inphase, whilst if the phase difference of two waves is (k2 + ), where k is an integer, they areconsidered in antiphase.

Multiples of radian units

Metric prefixeshave limited use with radians, and none in mathematics.

An approximation of the milliradian (0.001 rad), known as themil, is used ingunneryand

targeting. Based upon an approximation of = 3.2, there are 6400 mils in a complete rotation.

Other gunnery systems may use a different approximation to . Being based on the milliradian, it

corresponds roughly to an error of 1 m at a range of 1000 m (at such small angles, the curvature

is negligible). Thedivergenceoflaserbeams is also usually measured in milliradians.

Smaller units like microradians (rads) and nanoradians (nrads) are used in astronomy, and canalso be used to measure the beam quality of lasers with ultra-low divergence. Similarly, the

prefixes smaller than milli- are potentially useful in measuring extremely small angles.

Criticism

Main article:Turn (geometry)

The widespread habit of expressing radians in terms of fractions of has been criticized because

in this way one essentially use as a unit. An angle of radians represents half a circle, whichappear to be a rather arbitrary choice of unit. Therefore various mathematicians and scientists

have been critical costume of expressing radians as fractions of , and have proposed to use other

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constants instead. A. Eagle proposed to have a constant representing one quarter-turn.[5]

A

constant representing 2 has been proposed because a circle is 2 radians around. Robert Palaisproposed to use a "pi with three legs" ( ) to denote 1 turn,[6]while several people independently

have proposed to use the Greek letter (tau) instead.[7][8]

See also

http://en.wikipedia.org/wiki/Radian#cite_note-4http://en.wikipedia.org/wiki/Radian#cite_note-4http://en.wikipedia.org/wiki/Radian#cite_note-4http://en.wikipedia.org/wiki/Radian#cite_note-5http://en.wikipedia.org/wiki/Radian#cite_note-5http://en.wikipedia.org/wiki/Radian#cite_note-5http://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Radian#cite_note-6http://en.wikipedia.org/wiki/Radian#cite_note-6http://en.wikipedia.org/wiki/Radian#cite_note-6http://en.wikipedia.org/wiki/Radian#cite_note-6http://en.wikipedia.org/wiki/Radian#cite_note-6http://en.wikipedia.org/wiki/Tauhttp://en.wikipedia.org/wiki/Radian#cite_note-5http://en.wikipedia.org/wiki/Radian#cite_note-4