Radian Angle System

Radian Angle System

Radians measure angles as the ratio of the angle’s arc length over the radius of the circle used to draw the arc. 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.

Usage

The radian system is used for measuring angles and as the unit of choice for 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].

Definition

Radian Defintion

Radians are a unit that measures angle as the ratio of the angle’s arc length over the radius of a circle. The (equivalent) symbol is used to represent the radius-invariant property of the definiton[2]. A full rotation is equal to (tau) radians where is the naturally occurring circle constant defined by a circle’s circumference divided by its radius.

Measuring Angles

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.

1 over 12 tau radians
1 over 8 tau radians
1 over 6 tau radians
1 over 4 tau radians

Trigonometric Functions

Radians are often introduced with the trigonometric functions and the unit circle. Shown below are the plots of sine and cosine labeled in radians.

Graph of the cosine function labeled with radians. Graph of the sine function labeled with radians.

Radians vs Degrees

Radian Angle System
Degree Angle System

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.

References

  1. Radians Versus Degrees
    Wumbo (internal)
  2. Radians
    Wumbo (internal)
  3. No, really, pi is wrong: The Tau Manifesto
    Michael Hartl