Radian

The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at OEIS: A072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. Separately, the SI unit of solid angle measurement is the steradian. The radian is most commonly represented by the symbol rad. An alternative symbol is c, the superscript letter c (for 'circular measure'), the letter r, or a superscript R, but these symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r). So, for example, a value of 1.2 radians could be written as 1.2 rad, 1.2 r, 1.2rad, 1.2c, or 1.2R. Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length 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. As the ratio of two lengths, the radian is a 'pure number' that needs no unit symbol, and in mathematical writing the symbol 'rad' is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used. 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 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees. The relation 2 π  rad = 360 ∘ {displaystyle 2pi { ext{ rad}}=360^{circ }} can be derived using the formula for arc length. Taking the formula for arc length, or ℓ a r c = 2 π r ( θ 360 ∘ ) {displaystyle ell _{arc}=2pi rleft({frac { heta }{360^{circ }}} ight)} . Assuming a unit circle; the radius is therefore one. Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that 1 = 2 π ( 1  rad 360 ∘ ) {displaystyle 1=2pi left({frac {1{ ext{ rad}}}{360^{circ }}} ight)} . This can be further simplified to 1 = 2 π  rad 360 ∘ {displaystyle 1={frac {2pi { ext{ rad}}}{360^{circ }}}} . Multiplying both sides by 360 ∘ {displaystyle 360^{circ }} gives 360 ∘ = 2 π  rad {displaystyle 360^{circ }=2pi { ext{ rad}}} . The concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure. Prior to the term radian becoming widespread, the unit was commonly called circular measure of an angle.