Radian
An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.
General information
Unit system	SI
Unit of	angle
Symbol	rad, R[1]
Conversions
1 rad in ...	... is equal to ...
milliradians	1000 mrad
turns	1/2π turn
degrees	180/π° ≈ 57.296°
gradians	200/π grad ≈ 63.662g

The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius.^[2] The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit,^[2] defined in the SI as 1 rad = 1^[3] and expressed in terms of the SI base unit metre (m) as rad = m/m.^[4] Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.^[5]

Definition

One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle.^[6] More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, ${\displaystyle \theta ={\frac {s}{r}}}$, where θ is the subtended angle in radians, s is arc length, and r is radius. A right angle is exactly ${\displaystyle {\frac {\pi }{2}}}$ radians.^[7]

The rotation angle (360°) corresponding to one complete revolution is the length of the circumference divided by the radius, which is ${\displaystyle {\frac {2\pi r}{r}}}$, or 2π. Thus, 2π radians is equal to 360 degrees.

The relation 2π rad = 360° can be derived using the formula for arc length, ${\textstyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}$. Since radian is the measure of an angle that is subtended by an arc of a length equal to the radius of the circle, ${\textstyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}$. This can be further simplified to ${\textstyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}$. Multiplying both sides by 360° gives 360° = 2π rad.

Unit symbol

The International Bureau of Weights and Measures^[7] and International Organization for Standardization^[8] specify rad as the symbol for the radian. Alternative symbols that were in use in 1909 are ^c (the superscript letter c, for "circular measure"), the letter r, or a superscript ^R,^[1] but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence an angle of 1.2 radians would be written today as 1.2 rad; archaic notations could include 1.2 r, 1.2^rad, 1.2^c, or 1.2^R.

In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used.

Dimensional analysis

Plane angle may be defined as θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius. One radian corresponds to the angle for which s = r, hence 1 radian = 1 m/m.^[9] However, rad is only to be used to express angles, not to express ratios of lengths in general.^[7] A similar calculation using the area of a circular sector θ = 2A/r² gives 1 radian as 1 m²/m².^[10] The key fact is that the radian is a dimensionless unit equal to 1. In SI 2019, the radian is defined accordingly as 1 rad = 1.^[11] It is a long-established practice in mathematics and across all areas of science to make use of rad = 1.^[4][12]

Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations".^[13] For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimeters, where r is the radius of the pulley in centimeters and θ is the angle the pulley turns in radians. When multiplying r by θ the unit of radians disappears from the result. Similarly in the formula for the angular velocity of a rolling wheel, ω = v/r, radians appear in the units of ω but not on the right hand side.^[14] Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".^[15] Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".^[16]

In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s²), and torsional stiffness (N⋅m/rad), and not in the quantities of torque (N⋅m) and angular momentum (kg⋅m²/s).^[17]

At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle".^[18][19][20] Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the area of a circle, πr². The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".^[21] A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use.^[20]

In particular, Quincey identifies Torrens' proposal to introduce a constant η equal to 1 inverse radian (1 rad⁻¹) in a fashion similar to the introduction of the constant ε₀.^[21][a] With this change the formula for the angle subtended at the center of a circle, s = rθ, is modified to become s = ηrθ, and the Taylor series for the sine of an angle θ becomes:^[20][22]

where ${\displaystyle x=\eta \theta =\theta /{\text{rad}}}$. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed,^[22] while sin_rad is the traditional function on pure numbers which assumes its argument is in radians.^[23] ${\displaystyle \operatorname {Sin} }$ can be denoted ${\displaystyle \sin }$ if it is clear that the complete form is meant.^[20][24]

Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1. This radian convention allows the omission of η in mathematical formulas.^[25]

Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal.^[26] For example, the Boost units library defines angle units with a plane_angle dimension,^[27] and Mathematica's unit system similarly considers angles to have an angle dimension.^[28][29]

Conversions

Conversion of common angles

Turns	Radians	Degrees	Gradians
0 turn	0 rad	0°	0g
1/72 turn	π/36 rad	5°	5+5/9g
1/24 turn	π/12 rad	15°	16+2/3g
1/16 turn	π/8 rad	22.5°	25g
1/12 turn	π/6 rad	30°	33+1/3g
1/10 turn	π/5 rad	36°	40g
1/8 turn	π/4 rad	45°	50g
1/2π or τ turn	1 rad	approx. 57.3°	approx. 63.7g
1/6 turn	π/3 rad	60°	66+2/3g
1/5 turn	2π or τ/5 rad	72°	80g
1/4 turn	π/2 rad	90°	100g
1/3 turn	2π or τ/3 rad	120°	133+1/3g
2/5 turn	4π or α/5 rad	144°	160g
1/2 turn	π rad	180°	200g
3/4 turn	3π or ρ/2 rad	270°	300g
1 turn	τ or 2π rad	360°	400g

Between degrees

As stated, one radian is equal to ${\displaystyle {180^{\circ }}/{\pi }}$. Thus, to convert from radians to degrees, multiply by ${\displaystyle {180^{\circ }}/{\pi }}$.

${\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}$

For example:

${\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$

${\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}$

${\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}$Zdroj:https://en.wikipedia.org?pojem=Radians
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