Radians are a unit that measure angle as the ratio of the angle’s arclength over the radius of a circle. A full rotation in radians is equal to (tau) radians.
The radian system is used for measuring angles and as the unit of choice for the trigonometric functions. While the degree angle system is often used to introduce concepts, the radian system eventually becomes the preferred unit for measuring angles in math^{[1]}.
Radians are a unit that measure angle as the ratio of the angle’s arclength over the radius of a circle. The (equivalent) symbol is used to represent the radiusinvariant property of the definiton^{[2]}. A full rotation is equal to (tau) radians where is the naturally occuring circle constant defined by a circle’s circumference divided by its radius.
Angles measured using radians are usually expressed using the circle constant (tau). Shown below are some examples of angles measured using radians. The variable (theta) is a variable commonly used for angles. By convention, angles in the coordinate plane are measured from the positive direction where the counterclockwise rotation is positive.
The trigonometric functions and the unit circle are often where radians are introduced. Shown below are the plots of sine and cosine labeled in radians.
The short answer for why radians are preferred to degrees is that the radian system leads to more succinct and elegant formulas throughout mathematics^{[3]}. For example, the derivative of the sine function is only true when the angle is expressed in radians.
See the long answer on this page.
The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.
The greek letter π (pi) is a geometric constant approximately equal to 3.1416. Its value is equal to the length of any circle's circumference divided by its diameter.
Degrees are a unit of measure for angles. A full rotation is equal to 360 degrees. In the XY Cartesian Coordinate System, degrees are measured starting from the rightmost edge of the circle.
There are six trigonometric functions that relate to the geometry of the righttriangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.
The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the righttriangle's adjacent side over its hypotenuse.
This page compares and contrasts the two systems of measuring angles in math: radians and degrees, and explains why radians is the preferred unit of measure for angles.

Radians Versus DegreesWumbo (internal)

RadiansWumbo (internal)

No, really, pi is wrong: The Tau ManifestoMichael Hartl