# What is the reference angle?

Finds the reference angle of any angle in degrees, radians or a multiple of π: the acute angle to the x-axis, with the coterminal angle, the quadrant, and the point on the unit circle.

- Page: https://www.acalculator.org/math/reference-angle-calculator
- JSON spec: https://www.acalculator.org/math/reference-angle-calculator.json
- Version: 4d26e8fb4442

## Default answer

Example with the default inputs (Angle in Degrees, Angle 225): The reference angle is 45°.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| u | Angle in | Degrees, radians, or a multiple of π radians typed as a fraction (5/3 means 5π/3). |
| a | Angle | The angle θ; a negative angle turns clockwise. |
| k | Angle ÷ π | The angle as a multiple of π, such as 5/3 for 5π/3 or -1/6 for −π/6. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| reference | Reference angle (°) | The acute angle between the terminal side and the x-axis, 0° to 90°. |
| radians | Reference angle (rad) | The same reference angle in radians. |
| exact | In terms of π | The reference angle as an exact multiple of π. |
| coterminal | Coterminal angle (°) | The angle from 0° to under 360° that ends in the same place. |
| quadrant | Quadrant | Where the angle’s terminal side lies. |
| x | Point x (cos θ) | The x value of the angle’s point on the unit circle. |
| y | Point y (sin θ) | The y value of the angle’s point on the unit circle. |
| refX | Reference point x | cos of the reference angle: the point mirrored into quadrant I. |
| refY | Reference point y | sin of the reference angle. |
| steps | Working | How the reference angle is found. |

## Method

Find the coterminal angle c from 0° to under 360°; the reference angle is c in quadrant I, 180° − c in II, c − 180° in III and 360° − c in IV (the same in radians with π and 2π).

## Assumptions

- An angle on an axis has reference angle 0° (the x-axis) or 90° (the y-axis).
- Degrees and multiples of π are worked exactly; radians use π to double precision.
- Angles from −10⁹ to 10⁹ in degrees or radians, or −10¹² π to 10¹² π.

## Worked examples

1. u = deg, a = 225 gives reference = 45, exact = π/4, coterminal = 225, quadrant = Quadrant III, radians = 0.785398. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).
2. u = pi, k = 5/3 gives exact = π/3, reference = 60, coterminal = 300, quadrant = Quadrant IV. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).
3. u = pi, k = -1/6 gives exact = π/6, coterminal = 330, quadrant = Quadrant IV. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).
4. u = deg, a = 870 gives coterminal = 150, reference = 30, quadrant = Quadrant II, exact = π/6. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).
5. u = rad, a = 4 gives radians = 0.858407, quadrant = Quadrant III, reference = 49.183118. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).
6. u = deg, a = -90 gives coterminal = 270, reference = 90, quadrant = On the negative y-axis, exact = π/2. Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02).

## FAQ

### What is a reference angle?

The acute angle between the terminal side of an angle and the x-axis, from 0° to 90°. Angles with the same reference angle have the same sine and cosine up to the sign, so it lets you use the first-quadrant values for any angle.

### How do I find the reference angle?

First add or subtract 360° until the angle is from 0° to under 360°. Then use its quadrant: in quadrant I the angle is its own reference angle; in II use 180° − angle; in III use angle − 180°; in IV use 360° − angle. 225° is in quadrant III, so its reference angle is 225° − 180° = 45°.

### How do I find a reference angle in radians?

The same rules with π for 180° and 2π for 360°: π − t in quadrant II, t − π in III, 2π − t in IV. 5π/3 is in quadrant IV, so its reference angle is 2π − 5π/3 = π/3. Pick × π and type 5/3 to get the exact answer.

### What is the reference angle of a negative angle?

Add 360° (or 2π) first. −π/6 + 2π = 11π/6, which is in quadrant IV, so the reference angle is 2π − 11π/6 = π/6.

### What is the reference angle of an angle larger than 360°?

Subtract 360° as many times as needed. 870° − 720° = 150°, in quadrant II, so the reference angle is 180° − 150° = 30°.

### What is the reference angle of 90° or 180°?

An angle on an axis has no quadrant. The page uses the same rules at the boundaries, so an angle on the x-axis (0°, 180°) has reference angle 0°, and one on the y-axis (90°, 270°) has reference angle 90°.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (reference angles: the acute angle formed by the terminal side and the horizontal axis; quadrant rules; examples 225°, 315°, 5π/3, −π/6 and 7π/6). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)