Reference Angle Worksheet: Practice Problems with Answers

The 15 problems below cover every case: all 4 quadrants, radian measure, negative angles, and angles greater than 360°. Click "Show answer" to reveal the step-by-step solution for each.

Work through the problems in order without checking the answer first. The goal is to practice the same three-step method every time: normalize the angle, identify its quadrant, then apply the matching formula. Only open the answer once you've written down your own result, so you can confirm whether your reasoning matches the correct steps, not just the final number.

Formula reference card

Quadrant I

0° to 90°

θ̂ = θ

Quadrant II

90° to 180°

θ̂ = 180° − θ

Quadrant III

180° to 270°

θ̂ = θ − 180°

Quadrant IV

270° to 360°

θ̂ = 360° − θ

Keep this card visible while working through the problems below. Every answer on this worksheet comes directly from one of these four formulas. The only extra work needed is normalizing angles that start out negative or larger than 360° before checking which quadrant they land in.

Practice problems

Problem 1Find the reference angle for 60°.
  1. Already between 0° and 360°, Quadrant I
  2. θ̂ = 60°

Reference angle = 60°

Problem 2Find the reference angle for 120°.
  1. Already between 0° and 360°, Quadrant II
  2. θ̂ = 180° − 120° = 60°

Reference angle = 60°

Problem 3Find the reference angle for 200°.
  1. Already between 0° and 360°, Quadrant III
  2. θ̂ = 200° − 180° = 20°

Reference angle = 20°

Problem 4Find the reference angle for 310°.
  1. Already between 0° and 360°, Quadrant IV
  2. θ̂ = 360° − 310° = 50°

Reference angle = 50°

Problem 5Find the reference angle for 150°.
  1. Already between 0° and 360°, Quadrant II
  2. θ̂ = 180° − 150° = 30°

Reference angle = 30°

Problem 6Find the reference angle for 225°.
  1. Already between 0° and 360°, Quadrant III
  2. θ̂ = 225° − 180° = 45°

Reference angle = 45°

Problem 7Find the reference angle for 300°.
  1. Already between 0° and 360°, Quadrant IV
  2. θ̂ = 360° − 300° = 60°

Reference angle = 60°

Problem 8Find the reference angle for −90°.
  1. Normalize: −90° + 360° = 270°
  2. 270° is a quadrantal angle on the negative y-axis

Reference angle = 90°

Problem 9Find the reference angle for −150°.
  1. Normalize: −150° + 360° = 210°, Quadrant III
  2. θ̂ = 210° − 180° = 30°

Reference angle = 30°

Problem 10Find the reference angle for 390°.
  1. Normalize: 390° − 360° = 30°, Quadrant I
  2. θ̂ = 30°

Reference angle = 30°

Problem 11Find the reference angle for 540°.
  1. Normalize: 540° − 360° = 180°
  2. 180° is a quadrantal angle on the negative x-axis

Reference angle = 0°

Problem 12Find the reference angle for π/4.
  1. Already between 0 and 2π, Quadrant I
  2. θ̂ = π/4

Reference angle = π/4

Problem 13Find the reference angle for 3π/4.
  1. Already between 0 and 2π, Quadrant II
  2. θ̂ = π − 3π/4 = π/4

Reference angle = π/4

Problem 14Find the reference angle for 5π/6.
  1. Already between 0 and 2π, Quadrant II
  2. θ̂ = π − 5π/6 = π/6

Reference angle = π/6

Problem 15Find the reference angle for 7π/4.
  1. Already between 0 and 2π, Quadrant IV
  2. θ̂ = 2π − 7π/4 = π/4

Reference angle = π/4

Problems 8, 9, and 10 are worth revisiting if any of them gave trouble. They combine a negative angle or an angle over 360° with the quadrant formula in one step, which is where most arithmetic slips happen. Problems 12 through 15 do the same thing in radians instead of degrees, which is good practice for recognizing that the method never actually changes between the two units, only the numbers π and 2π replace 180° and 360°.

If a problem's answer doesn't match, work back through the three steps individually rather than redoing the whole problem at once. Check the normalized angle first, then the quadrant, then the formula. Almost every mistake on this worksheet traces back to one of those three steps, not to the final subtraction itself.

Frequently Asked Questions

How do you find a reference angle step by step?

Normalize the angle between 0° and 360° (or 0 and 2π), identify its quadrant, then apply that quadrant's formula: θ, 180°−θ, θ−180°, or 360°−θ. These same three steps solve every problem on this worksheet, regardless of the starting angle or its unit.

What is the reference angle for 200°?

The reference angle for 200° is 20°. 200° is in Quadrant III, so θ̂ = 200° − 180° = 20°. Tangent alone stays positive at 200°, since it is the only trig function that remains positive throughout all of Quadrant III.

What is the reference angle for 310°?

The reference angle for 310° is 50°. 310° is in Quadrant IV, so θ̂ = 360° − 310° = 50°. Cosine alone stays positive at 310°, since it is the only trig function that remains positive throughout all of Quadrant IV.

What is the reference angle for 3π/4?

The reference angle for 3π/4 is π/4. 3π/4 is in Quadrant II, so θ̂ = π − 3π/4 = π/4. Sine alone stays positive at 3π/4, since it is the only trig function that remains positive throughout all of Quadrant II.

What is the reference angle for 7π/4?

The reference angle for 7π/4 is π/4. 7π/4 is in Quadrant IV, so θ̂ = 2π − 7π/4 = π/4. Cosine still stays positive at 7π/4 as well, matching the same Quadrant IV sign pattern already used for 310° above.