Radian Angle System

This figure illustrates the radian angle system annotated using the circle constant τ (tau).
Figure 1: Radian Angle System

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.

Usage

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].

Definition

This figure illustrates the geoemtric definiton of a radian angle.
Figure 2: Radians Definition

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 represent the radius-invariant 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.

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.

This figure illustrates the standardized angle that represents an twelfth turn.
Figure 3: 1/12 τ Radians
This figure illustrates the standardized angle that represents an eighth turn.
Figure 4: 1/8 τ Radians
This figure illustrates the standardized angle that represents an sixth turn.
Figure 5: 1/6 τ Radians
This figure illustrates the standardized angle that represents an fourth turn.
Figure 6: 1/4 τ Radians

Trigonometric Functions

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.

Plot of Cosine from 0 to τ (tau) radians.
Figure 7: Cosine Plot
Plot of Sine Function from 0 to τ (tau) radians.
Figure 8: Sine Plot

Radians vs Degrees

This figure illustrates the radian angle system annotated using the circle constant τ (tau).
Figure 9: Radian Angle System
This figure illustrates the degree angle system which represents a full rotation as 360 units called degrees.
Figure 10: 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.

Links

References

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