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One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals 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, =, where θ is the magnitude in radians of the subtended angle, s is arc length, and r is radius.
The radian is determined by the circumference of a circle that is equal in length to the radius of the circle (n = 2 π = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is rad. One turn is 2 π radians, and one radian is 180° / π , or
The radian per second (symbol: rad⋅s −1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency (symbol ω, omega). The radian per second is defined as the angular frequency that results in the angular displacement increasing by one radian every ...
Additionally, an angle that is a rational multiple of radians is constructible if and only if, when it is expressed as / radians, where a and b are relatively prime integers, the prime factorization of the denominator, b, is the product of some power of two and any number of distinct Fermat primes (a Fermat prime is a prime number one greater ...
provided the angle is measured in radians. Angles measured in degrees must first be converted to radians by multiplying them by / . These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science.
An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which is expressed here using the Greek letter tau (τ). Some special angles in radians, stated in terms of 𝜏. A comparison of angles expressed in degrees and radians.
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
Rotational speed is sometimes used to mean angular frequency rather than the quantity defined in this article. Angular frequency gives the change in angle per time unit, which is given with the unit radian per second in the SI system.