Radian
The radian is a unit of plane
angle. It is represented by the symbol "rad" or, more rarely, by the superscript c . For example, an angle of 1.2 radians would be written "1.2 rad" or "1.2c ".
The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995
and the radian is now considered an SI derived unit. For measuring solid angles, see
steradian.
Nowadays, radian is the de facto unit of plane angles for mathematicians, and the symbol "rad" is usually omitted in mathematicial writings. When using degrees, the symbol is used to distinguish it from radians.
Encyclopedia
The
radian is a unit of plane
angle. It is represented by the symbol "rad" or, more rarely, by the superscript c . For example, an angle of 1.2 radians would be written "1.2 rad" or "1.2
c ".
The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995
and the radian is now considered an SI derived unit. For measuring solid angles, see
steradian.
Nowadays, radian is the de facto unit of plane angles for mathematicians, and the symbol "rad" is usually omitted in mathematicial writings. When using degrees, the ° symbol is used to distinguish it from radians.
Definition
One radian is the angle subtended at the center of a
circle by an arc of circumference that is equal in length to the radius of the circle.
In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle.
History
The term
radian first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College,
Belfast. James Thomson was a brother of
Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of
St. Andrew's University, hesitated between
rad,
radial and
radian. In 1874, Muir adopted
radian after a consultation with James Thomson. .
The concept of a 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.
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 . Retaining the word radian emphasizes that angular velocity is equal to 2p 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.
There are 2
p radians in a complete circle, so:
or:
More generally, we can say:
If, for example,
-1.570796 in radians was given, the corresponding degree value would be:
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
,
which is the basis of many other elegant identities in mathematics, including
.
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 sin
x:
If
x had units, then the sum would be meaningless; the linear term
x cannot be added to the cubic term , etc. Therefore,
x must be dimensionless.
SI multiples
SI prefixes have limited use with radians. The milliradian is used in
gunnery and general
targeting, because it corresponds to 1 m at a range of 1000 m. 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 2p radians is to begin the same circle again.
See also
...
External links
