Radians and Degrees

Angle measure, full turns, coterminal angles, and reference angles.

Angles measure rotation. The two most common units are degrees and radians. Degrees are familiar from geometry: a full turn is 360 degrees. Radians are the natural angle unit for advanced mathematics: a full turn is 2pi radians.

Learning objectives

After this lesson you should be able to:

  • explain degrees and radians as two angle units;
  • convert degrees to radians and radians to degrees;
  • find coterminal angles;
  • find reference angles;
  • avoid mixing degree-mode and radian-mode calculations;
  • check conversions with edumath and SymPy.

Degrees

A full turn is divided into 360 degrees.

full turn = 360 degrees
half turn = 180 degrees
quarter turn = 90 degrees

Degrees are useful for drawing and interpreting everyday angles.

Radians

Radians measure angle by comparing arc length to radius:

radian measure = arc length / radius

On a unit circle, the radius is 1, so the radian measure equals the arc length. A full circle has circumference 2pi, so:

360 degrees = 2pi radians
180 degrees = pi radians
90 degrees = pi/2 radians

Conversion formulas

degrees to radians: multiply by pi/180
radians to degrees: multiply by 180/pi

Example:

60 degrees = 60 * pi/180 = pi/3 radians

Coterminal angles

Coterminal angles land in the same direction because they differ by full turns.

30 degrees, 390 degrees, and -330 degrees

are coterminal because they differ by multiples of 360 degrees.

Reference angles

A reference angle is the acute angle between the terminal side and the x-axis. It helps you use special-angle values in any quadrant.

Examples:

150 degrees has reference angle 30 degrees
225 degrees has reference angle 45 degrees
300 degrees has reference angle 60 degrees

Worked example 1: convert degrees to radians

Convert 135 degrees to radians.

135 degrees * pi/180 = 135pi/180 = 3pi/4

So 135 degrees = 3pi/4 radians.

Worked example 2: coterminal angle

Find a coterminal angle between 0 and 360 degrees for -75 degrees.

Add 360:

-75 + 360 = 285

So 285 degrees is coterminal with -75 degrees.

Worked example 3: reference angle

Find the reference angle for 210 degrees.

  1. 210 degrees is in Quadrant III.
  2. In Quadrant III, subtract 180 degrees.
  3. 210 - 180 = 30.

The reference angle is 30 degrees.

Common pitfalls

  • Forgetting that pi radians is 180 degrees, not 360 degrees.
  • Using degree values in a calculator set to radian mode.
  • Thinking coterminal angles are different directions.
  • Forgetting to reduce negative angles by adding full turns.

Practice exercises

  1. Convert 90 degrees to radians.
  2. Convert pi/6 radians to degrees.
  3. Find a coterminal angle in [0,360) for -45 degrees.
  4. Find the reference angle for 240 degrees.
  5. Which is larger: pi/2 radians or 60 degrees?
  1. 90*pi/180 = pi/2.
  2. (pi/6)*(180/pi)=30 degrees.
  3. -45+360=315 degrees.
  4. 240-180=60 degrees.
  5. pi/2 radians = 90 degrees, so it is larger than 60 degrees.

Subtopic guided practice and checkpoints

Angle problems are safest when you identify the unit before doing arithmetic.

Lab 1: convert degrees to radians

Guess first. Should 30 degrees be smaller or larger than pi radians?

Guided exercise.

30 degrees * pi/180 = pi/6 radians

Since pi radians is a half turn, pi/6 is much smaller.

Checkpoint. Convert 120 degrees to radians.

Lab 2: convert radians to degrees

Guess first. Is 3pi/2 radians more than one full turn?

Guided exercise.

(3pi/2) * 180/pi = 270 degrees

A full turn is 360 degrees, so 270 degrees is three quarters of a turn.

Checkpoint. Convert 5pi/6 radians to degrees.

Lab 3: find coterminal angles

Guess first. What full turn can you add to a negative angle?

Guided exercise.

For -120 degrees:

-120 + 360 = 240 degrees

Both angles point in the same direction.

Checkpoint. Find a coterminal angle in [0,360) for 765 degrees.

Lab 4: find reference angles

Guess first. Is the reference angle always positive and acute?

Guided exercise.

For 330 degrees, the terminal side is in Quadrant IV. The angle to the positive x-axis is:

360 - 330 = 30 degrees

Checkpoint. Find the reference angle for 135 degrees.

  1. 120*pi/180 = 2pi/3.
  2. (5pi/6)*(180/pi)=150 degrees.
  3. 765-720=45 degrees.
  4. 180-135=45 degrees.

Angle-measure guessing game

Using this lesson with edumath and SymPy

from edumath.trigonometry import (
    coterminal_angle,
    degrees_to_radians,
    radians_to_degrees,
    reference_angle_degrees,
)

print(degrees_to_radians(60))
print(radians_to_degrees(3.141592653589793 / 2))
print(coterminal_angle(-75))
print(reference_angle_degrees(210))
1.0471975511965976
90.0
285
30
import sympy as sp

sp.simplify(135 * sp.pi / 180), sp.simplify((5 * sp.pi / 6) * 180 / sp.pi)
(3*pi/4, 150)