Quick answer

Let θ be in [0°, 360°) or [0, 2π). Then α follows the quadrant:

Formula

  • QI: α = θ
  • QII: α = 180° − θ (π − θ)
  • QIII: α = θ − 180° (θ − π)
  • QIV: α = 360° − θ (2π − θ)

Introduction

These four lines are the core of reference angle work. Pair them with the Reference Angle Calculator so unit switches between degrees, radians, and π multiples stay consistent.

Each formula answers the same question: how far is the terminal side from the nearest x-axis direction, measured as an acute angle? The quadrant tells you which baseline to subtract from.

If you need the definition and the 0° to 90° constraint explained in plain language, read what is a reference angle before you memorize these lines.

Students who rush to subtract often pick the wrong line for Quadrant II or IV. A quick sketch beats memorizing mnemonics that do not say which axis you are measuring from.

Why each quadrant differs

The acute gap is measured from different sides of the x-axis. In Quadrant II, θ is measured counterclockwise from the positive x-axis, but α is the small angle between the terminal side and the negative x-axis, which is why α = 180° − θ.

In Quadrant III, both the positive and negative x directions are nearby, but the standard rule subtracts π or 180° because the terminal side lies past the negative x-axis. You are measuring how far θ extends beyond half a turn.

Quadrant IV uses 360° − θ or 2π − θ because θ is short of a full rotation and α is the leftover gap back to the positive x-axis. The structure repeats in radians with π and 2π.

Boundary angles need care: 90°, 180°, 270°, and 360° sit on axes. Confirm whether your course treats α as 0° or 90° when the terminal side is vertical.

Degree and radian forms

  • Degrees: use 180° and 360°
  • Radians: use π and 2π
  • Constraint: 0° ≤ α ≤ 90°
  • QI: α = θ when θ is between 0° and 90°

Example in QIV: θ = 315° gives α = 360° − 315° = 45°. Example in QIII: θ = 5π/4 gives α = 5π/4 − π = π/4. The arithmetic is short; the quadrant label is where mistakes happen.

When a problem states θ in radians but you think in degrees, convert once and keep the unit through the entire subtraction so you do not mix π with 180° in the same line.

If θ is negative, normalize first. −135° becomes 225° in standard positive rotation, which lands in QIII, so α = 225° − 180° = 45°, not 135° from a wrong quadrant guess.

Formula checklist

  1. Normalize θ. Add or subtract full turns until θ lies in [0°, 360°) or [0, 2π). This step is shared with coterminal-angle work, but here you stop after one representative turn.
  2. Name the quadrant. Compare θ to 90°, 180°, and 270° (or π/2, π, 3π/2). Write QI, QII, QIII, or QIV on your diagram before you choose a formula line.
  3. Subtract with the correct line. Apply exactly one of the four quick-answer formulas. If the result is not acute, you used the wrong quadrant or forgot to normalize.
  4. Verify on the calculator. Match α and the quadrant label to the home tool diagram. Disagreement usually means a normalization error, not a broken formula.

Example: 7π/6

7π/6 radians is 210°, which lies in Quadrant III. The reference formula for QIII gives α = 7π/6 − π = π/6, or 30°.

Check the acute constraint: π/6 is between 0 and π/2. If you had accidentally used π − 7π/6, you would get a negative value, which signals the wrong quadrant rule.

Enter 1.167 with π rad selected on the home calculator (7/6 ≈ 1.167) to verify the quadrant text and the dashed reference arc. For six more quadrant samples, see reference angle examples.