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Coterminal Angles

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

Two angles are coterminal when they share the same terminal ray, which happens exactly when they differ by a whole number of full rotations: 360 degrees (or 2pi radians). To find a coterminal angle you add or subtract 360 degrees (or 2pi) as many times as needed; to land in [0, 360) you keep adjusting until the result is at least 0 and less than 360. Two angles are coterminal if and only if their difference is an integer multiple of 360 degrees (or 2pi).

What you'll learn

Why it matters: A clock hand pointing to the 3 looks identical whether it has spun a quarter turn or a full extra loop past it, and a compass bearing of 380 degrees aims exactly where 20 degrees does. Coterminal angles describe these "same direction, different total spin" situations.

Worked example

Problem. Find the angle in [0, 360) that is coterminal with 765 degrees. Then give one positive and one negative coterminal angle.

  1. Subtract 360 to shrink toward the range: 765 - 360 = 405, still 360 or more, so subtract again: 405 - 360 = 45. Since 0 <= 45 < 360, the in-range coterminal angle is 45 degrees.
  2. Add 360 to 45 for another positive coterminal angle: 45 + 360 = 405 degrees.
  3. Subtract 360 from 45 for a negative coterminal angle: 45 - 360 = -315 degrees.

Answer: In-range: 45 degrees. A positive coterminal angle: 405 degrees. A negative coterminal angle: -315 degrees.

Practice problems

1. Find the angle in [0, 360) that is coterminal with 405 degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the angle in [0, 360) that is coterminal with 405 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 405 is at least 360, so subtract one full rotation: 405 - 360 = 45.
  4. Since 0 <= 45 < 360, the in-range coterminal angle is 45 degrees.
  5. Check the result by substituting or estimating: the response should match 45 degrees and make sense in the original problem.

Answer: 45 degrees

2. Find the angle in [0, 360) that is coterminal with -30 degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the angle in [0, 360) that is coterminal with -30 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. -30 is negative, so add one full rotation: -30 + 360 = 330.
  4. Since 0 <= 330 < 360, the in-range coterminal angle is 330 degrees.
  5. Check the result by substituting or estimating: the response should match 330 degrees and make sense in the original problem.

Answer: 330 degrees

3. Which angle is coterminal with 90 degrees?

Choices: 180 degrees · 450 degrees · 270 degrees · -90 degrees

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which angle is coterminal with 90 degrees?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. A coterminal angle differs from 90 by a multiple of 360.
  4. 450 - 90 = 360, which is one full rotation, so 450 degrees is coterminal with 90 degrees.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 450 degrees

4. Find the angle in [0, 360) that is coterminal with 810 degrees.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the angle in [0, 360) that is coterminal with 810 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Subtract full rotations: 810 - 360 = 450, still too big, so 450 - 360 = 90.
  4. Since 0 <= 90 < 360, the in-range coterminal angle is 90 degrees.
  5. Check the result by substituting or estimating: the response should match 90 degrees and make sense in the original problem.

Answer: 90 degrees

5. Find the angle in [0, 360) that is coterminal with -540 degrees.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the angle in [0, 360) that is coterminal with -540 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Add full rotations: -540 + 360 = -180, still negative, so -180 + 360 = 180.
  4. Since 0 <= 180 < 360, the in-range coterminal angle is 180 degrees.
  5. Check the result by substituting or estimating: the response should match 180 degrees and make sense in the original problem.

Answer: 180 degrees

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