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Reference Angles

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

A reference angle is the positive acute angle formed between the terminal side of an angle and the x-axis (never the y-axis). Which rule you use depends on the quadrant: in Quadrant I the reference angle equals the angle itself; in Quadrant II it is 180-theta (pi-theta); in Quadrant III it is theta-180 (theta-pi); and in Quadrant IV it is 360-theta (2pi-theta). The result is always between 0 and 90 degrees (0 and pi/2 radians).

What you'll learn

Why it matters: Reference angles let engineers and physicists reduce any direction of a rotating part, wave, or force vector back to a single first-quadrant angle, so one known trig value can be reused for every quadrant instead of memorizing values all the way around the circle.

Worked example

Problem. Find the reference angle for 225 degrees.

  1. 225 degrees is between 180 and 270, so its terminal side is in Quadrant III.
  2. In Quadrant III the reference angle is theta - 180 = 225 - 180.
  3. 225 - 180 = 45, so the reference angle is 45 degrees.

Answer: 45 degrees

Practice problems

1. Find the reference angle for 150 degrees.

Show solution
  1. 150 degrees lies between 90 and 180, so it is in Quadrant II.
  2. In Quadrant II the reference angle is 180 - theta = 180 - 150.
  3. 180 - 150 = 30, so the reference angle is 30 degrees.

Answer: 30 degrees

2. Find the reference angle for 300 degrees.

Show solution
  1. 300 degrees lies between 270 and 360, so it is in Quadrant IV.
  2. In Quadrant IV the reference angle is 360 - theta = 360 - 300.
  3. 360 - 300 = 60, so the reference angle is 60 degrees.

Answer: 60 degrees

3. Which expression gives the reference angle for an angle theta whose terminal side is in Quadrant III (between 180 and 270 degrees)?

Choices: 180 - theta · theta - 180 · 360 - theta · theta

Show solution
  1. In Quadrant III the terminal side has rotated past 180 degrees.
  2. The reference angle is the gap back to the negative x-axis at 180 degrees.
  3. That gap is theta - 180.

Answer: theta - 180

4. Find the reference angle for 120 degrees.

Show solution
  1. 120 degrees lies between 90 and 180, so it is in Quadrant II.
  2. In Quadrant II the reference angle is 180 - theta = 180 - 120.
  3. 180 - 120 = 60, so the reference angle is 60 degrees.

Answer: 60 degrees

5. Find the reference angle for 210 degrees.

Show solution
  1. 210 degrees lies between 180 and 270, so it is in Quadrant III.
  2. In Quadrant III the reference angle is theta - 180 = 210 - 180.
  3. 210 - 180 = 30, so the reference angle is 30 degrees.

Answer: 30 degrees

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