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Reference angle

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Reference Angle

The reference angle is the acute angle between the terminal side of an angle in standard position and the x-axis. Learn the per-quadrant formula and how to find reference angles in degrees and radians, with examples.

What a reference angle is

The reference angle is the acute angle — always between 0° and 90° — formed between the terminal side of an angle in standard position and the x-axis. Every angle has a reference angle, and it is what lets you connect any angle back to the first quadrant, where the trig ratios are easiest to evaluate. The reference angle is always measured to the x-axis, never the y-axis, and it is always positive.

How to find the reference angle in each quadrant

Reference angle in Quadrant II xy terminal side θ ref. angle θ' Q IIQ I
The reference angle is the acute angle between the terminal side and the x-axis.

For an angle θ in standard position (measured from the positive x-axis, between 0° and 360°), the reference angle depends on the quadrant of the terminal side:

  • Quadrant I (0° to 90°): reference angle = θ
  • Quadrant II (90° to 180°): reference angle = 180° - θ
  • Quadrant III (180° to 270°): reference angle = θ - 180°
  • Quadrant IV (270° to 360°): reference angle = 360° - θ

Knowing the quadrant also tells you the sign of each trig ratio, which is where the ASTC rule comes in.

Examples in degrees

  • 150° is in Quadrant II: reference angle = 180° - 150° = 30°.
  • 210° is in Quadrant III: reference angle = 210° - 180° = 30°.
  • 320° is in Quadrant IV: reference angle = 360° - 320° = 40°.

You can check the position of each angle on the unit circle.

Examples in radians

In radians, replace 180° with π and 360° with 2π:

  • 5π/6 is in Quadrant II: reference angle = π - 5π/6 = π/6.
  • 4π/3 is in Quadrant III: reference angle = 4π/3 - π = π/3.

If your angle is given in one unit and you need the other, see how to convert between degrees and radians.

Reference angle vs coterminal angle

These are easy to mix up. A coterminal angle shares the same terminal side (you get one by adding or subtracting 360°), so it can be any size. A reference angle is always the acute angle to the x-axis. If an angle is larger than 360° or negative, first find a coterminal angle between 0° and 360°, then take its reference angle.

Common mistakes

  • Measuring to the y-axis. The reference angle is always taken to the x-axis.
  • Using the wrong quadrant formula. Identify the quadrant first, then apply that quadrant's rule.
  • Leaving it negative or larger than 90°. A reference angle is always acute and positive.

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