Everything about The Radian totally explained
The
radian is a unit of plane
angle, equal to 180/
π degrees, or about 57.2958 degrees. It is the standard unit of angular measurement in all areas of
mathematics beyond the elementary level.
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 can be mistaken for a degree: "1.2°"). However, the radian is mathematically considered a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol
° is used.
The radian was formerly an
SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an
SI derived unit. The SI unit of
solid angle measurement is the
steradian.
Definition
One radian is the
angle subtended at the center of a
circle by an
arc that's equal in length to the
radius of the circle.
More generally, the magnitude in radians of any angle subtended by two radii is equal to the ratio of the length of the enclosed arc to the radius of the circle; that is,
θ =
s /
r, where
θ is the subtended angle in radians,
s is arc length, and
r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is,
s =
rθ.
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 2π
r /
r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.
History
The concept of radian measure, as opposed to the degree of an angle, should probably be credited to
Roger Cotes in 1714. 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
June 5,
1873, in examination questions set by
James Thomson (brother of
Lord Kelvin) at
Queen's College,
Belfast. He used the term as early as 1871, while in 1869,
Thomas Muir, then of the
University of St Andrews, vacillated between
rad,
radial and
radian. In
1874, Muir adopted
radian after a consultation with James Thomson.
Conversions
Conversion between radians and degrees
As stated above, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,
»
If
x had units, then the sum would be meaningless: the linear term
x can't be added to (or have subtracted) the cubic term
or the quintic term
, etc. Therefore,
x must be dimensionless.
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 2π radians per second.
Similarly,
angular acceleration is often measured in radians per second per second (rad/s
2).
The reasons are the same as in mathematics.
Multiples of radian units
Metric prefixes have limited use with radians, and none in mathematics.
The milliradian (0.001 rad, or 1 mrad) is used in
gunnery and
targeting, because it corresponds to an error of 1 m at a range of 1000 m (at such small angles, the curvature is negligible).
The
divergence of
laser beams is also usually measured in milliradians.
Smaller units like microradians (μrads) and nanoradians (nrads) are used in astronomy, and can also 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.
However, the larger prefixes have no apparent utility, mainly because to exceed 2π radians is to begin the same circle (or revolutionary cycle) again.
Further Information
Get more info on 'Radian'.
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