Reference Angle Examples with Step-by-Step Solutions

Twelve worked examples covering every case: all four quadrants, negative angles, angles over 360°, and radians. Use the quick calculator below for any angle of your own.

Reading worked examples is most useful when you pause before each result and predict it yourself. The method never changes between examples, only the starting angle does, so the real skill being built here is recognizing which situation applies: is the angle already between 0° and 360°, does it need a sign flip first, or does it need reducing by a full turn before anything else?

1200° falls in Quadrant III
2θ̂ = 200° − 180°
3Reference angle = 20°

Every case, solved

Each example below follows the same three-step pattern: normalize the angle if needed, identify its quadrant, then apply that quadrant's formula. Working through all twelve in order builds familiarity with every situation a reference angle problem can present: positive angles in each of the four quadrants, negative angles, angles that wrap past 360°, and angles given in radians instead of degrees.

Example 1: Find the reference angle for 30°

  1. 30° is between 0° and 90°, Quadrant I
  2. θ̂ = 30°

Reference angle = 30°

Example 2: Find the reference angle for 135°

  1. 135° is between 90° and 180°, Quadrant II
  2. θ̂ = 180° − 135° = 45°

Reference angle = 45°

Example 3: Find the reference angle for 225°

  1. 225° is between 180° and 270°, Quadrant III
  2. θ̂ = 225° − 180° = 45°

Reference angle = 45°

Example 4: Find the reference angle for 315°

  1. 315° is between 270° and 360°, Quadrant IV
  2. θ̂ = 360° − 315° = 45°

Reference angle = 45°

Example 5: Find the reference angle for 150°

  1. 150° is between 90° and 180°, Quadrant II
  2. θ̂ = 180° − 150° = 30°

Reference angle = 30°

Example 6: Find the reference angle for 210°

  1. 210° is between 180° and 270°, Quadrant III
  2. θ̂ = 210° − 180° = 30°

Reference angle = 30°

Example 7: Find the reference angle for −45°

  1. Add 360°: −45° + 360° = 315°, Quadrant IV
  2. θ̂ = 360° − 315° = 45°

Reference angle = 45°

Example 8: Find the reference angle for −120°

  1. Add 360°: −120° + 360° = 240°, Quadrant III
  2. θ̂ = 240° − 180° = 60°

Reference angle = 60°

Example 9: Find the reference angle for 450°

  1. Subtract 360°: 450° − 360° = 90°
  2. 90° is a quadrantal angle on the positive y-axis

Reference angle = 90°

Example 10: Find the reference angle for 720°

  1. Subtract 360° twice: 720° − 360° − 360° = 0°
  2. 0° is a quadrantal angle on the positive x-axis

Reference angle = 0°

Example 11: Find the reference angle for 2π/3

  1. 2π/3 is between π/2 and π, Quadrant II
  2. θ̂ = π − 2π/3 = π/3

Reference angle = π/3

Example 12: Find the reference angle for 5π/4

  1. 5π/4 is between π and 3π/2, Quadrant III
  2. θ̂ = 5π/4 − π = π/4

Reference angle = π/4

Look back over the twelve results and a pattern emerges: only three distinct reference angles show up: 30°, 45°, and 60° (along with the quadrantal cases of 0° and 90°), even though the twelve starting angles look completely different from each other. That's the reference angle idea in action: a small set of first-quadrant values, reflected across all four quadrants, covers the vast majority of angles that appear in typical trigonometry problems.

Examples 7 through 10 are worth a second look, since they're the ones students most often get wrong on a first attempt. Negative angles and angles over 360° both require one extra step, adding or subtracting 360°, before the ordinary quadrant formula can be applied at all. Skipping that normalization step and plugging the original angle straight into a quadrant formula is the single most common mistake in this entire topic.

Examples 11 and 12 switch to radians specifically to show that nothing about the method changes with the unit. The same normalize-quadrant-formula sequence applies, just with π and 2π standing in for 180° and 360°. Anyone comfortable with examples 1 through 6 in degrees should find 11 and 12 straightforward, since the arithmetic is the only part that looks different.

Frequently Asked Questions

What is the reference angle for 135°?

The reference angle for 135° is 45°. 135° is in Quadrant II, so θ̂ = 180° − 135° = 45°. Since sine is positive in Quadrant II, sin(135°) equals sin(45°), while cosine and tangent are both negative at 135° instead.

What is the reference angle for 210°?

The reference angle for 210° is 30°. 210° is in Quadrant III, so θ̂ = 210° − 180° = 30°. In Quadrant III, tangent alone stays positive, so tan(210°) equals tan(30°), while both sine and cosine turn negative there instead.

What is the reference angle for −45°?

The reference angle for −45° is 45°. Adding 360° gives the positive coterminal angle 315°, which is in Quadrant IV, so θ̂ = 360° − 315° = 45°. Negative angles always convert to a positive coterminal angle first, before any quadrant formula is applied.

What is the reference angle for 450°?

The reference angle for 450° is 90°. Subtracting 360° gives 90°, which is a quadrantal angle on the positive y-axis. Any angle above 360° is reduced the same way, by subtracting full turns until the result falls between 0° and 360°.

What is the reference angle for 2π/3?

The reference angle for 2π/3 is π/3. 2π/3 is in Quadrant II, so θ̂ = π − 2π/3 = π/3. The same Quadrant II formula used for degrees, 180° minus the angle, applies here with π standing in for 180°.