Radians are a unit that measure angle as the ratio of the angle’s arc-length over the radius of a circle. A full rotation in radians is equal to (tau) radians.
Note: This website uses the constant (tau) instead of (pi) as the default circle constant. The substitution can be used to translate between the two constants.
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.
Radians are a unit that measure angle as the ratio of the angle’s arc-length over the radius of a circle. The (equivalent) symbol is used to indicate that negative angles or angles more than a full rotation are equivalent angle to an angle between and a full rotation. A full rotation is equal to (tau) radians.
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 counter-clockwise rotation is positive.
The short answer for why radians are preferred to degrees is that the radian system leads to more succinct and elegant formulas throughout mathematics. For example, the derivative of the sine function is only true when the angle is expressed in radians.
There are six trigonometric functions that relate to the geometry of the right-triangle 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.