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Degrees and Radians

A free Trigonometry lesson from the “Angles, Degrees, and Radians” unit, with a worked example and practice problems including step-by-step solutions.

Degrees and radians are two units for measuring the same angle, connected by the fact that 180 degrees equals pi radians. To go from degrees to radians, multiply by pi/180; to go from radians to degrees, multiply by 180/pi. Simplifying the resulting fraction gives the clean forms you'll use throughout trigonometry, such as 30 degrees = pi/6 and 3pi/4 = 135 degrees.

What you'll learn

Why it matters: Radians are the natural unit for rotation in physics and engineering, so software for robotics, animation, and GPS routinely converts between the degrees people read on a protractor or compass and the radians the math actually runs on.

Worked example

Problem. Convert 270 degrees to radians, and convert 5pi/6 radians to degrees.

  1. Degrees to radians: multiply by pi/180, so 270 * pi/180 = 270pi/180.
  2. Simplify 270/180 = 3/2, giving 3pi/2 radians.
  3. Radians to degrees: multiply by 180/pi, so (5pi/6)(180/pi) = 5*180/6 = 900/6 = 150 degrees.

Answer: 270 degrees = 3pi/2 radians; 5pi/6 radians = 150 degrees

Practice problems

1. Convert 180 degrees to radians.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert 180 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply by pi/180: 180 * pi/180.
  4. 180/180 = 1, so the result is pi radians.
  5. Check the result by substituting or estimating: the response should match pi and make sense in the original problem.

Answer: pi

2. Convert 90 degrees to radians.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert 90 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply by pi/180: 90 * pi/180 = 90pi/180.
  4. Simplify 90/180 = 1/2, giving pi/2 radians.
  5. Check the result by substituting or estimating: the response should match pi/2 and make sense in the original problem.

Answer: pi/2

3. Which factor converts a radian measure into degrees?

Choices: pi/180 · 180/pi · pi/360 · 360/pi

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which factor converts a radian measure into degrees?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Since 180 degrees = pi radians, multiplying by 180/pi cancels the pi and leaves degrees.
  4. Multiplying by pi/180 would go the other direction (degrees to radians).
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 180/pi

4. Convert 30 degrees to radians.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Convert 30 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply by pi/180: 30 * pi/180 = 30pi/180.
  4. Simplify 30/180 = 1/6, giving pi/6 radians.
  5. Check the result by substituting or estimating: the response should match pi/6 and make sense in the original problem.

Answer: pi/6

5. Convert 270 degrees to radians.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Convert 270 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Multiply by pi/180: 270 * pi/180 = 270pi/180.
  4. Simplify 270/180 = 3/2, giving 3pi/2 radians.
  5. Check the result by substituting or estimating: the response should match 3pi/2 and make sense in the original problem.

Answer: 3pi/2

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