{
  "id": "reference-angle",
  "version": "4d26e8fb4442",
  "status": "published",
  "name": "Reference Angle Calculator",
  "question": "What is the reference angle?",
  "summary": "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.",
  "category": "math",
  "subcategory": "trigonometry",
  "url": "https://www.acalculator.org/math/reference-angle-calculator",
  "markdown": "https://www.acalculator.org/math/reference-angle-calculator.md",
  "kind": "function",
  "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "u": {
        "title": "Angle in",
        "description": "Degrees, radians, or a multiple of π radians typed as a fraction (5/3 means 5π/3).",
        "type": "string",
        "enum": [
          "deg",
          "rad",
          "pi"
        ]
      },
      "a": {
        "title": "Angle",
        "description": "The angle θ; a negative angle turns clockwise.",
        "type": "number",
        "minimum": -1000000000,
        "maximum": 1000000000
      },
      "k": {
        "title": "Angle ÷ π",
        "description": "The angle as a multiple of π, such as 5/3 for 5π/3 or -1/6 for −π/6.",
        "type": "string",
        "pattern": "^-?\\d+( \\d+)?(/\\d+)?$"
      }
    }
  },
  "outputs": {
    "reference": {
      "label": "Reference angle (°)",
      "description": "The acute angle between the terminal side and the x-axis, 0° to 90°.",
      "format": "number"
    },
    "radians": {
      "label": "Reference angle (rad)",
      "description": "The same reference angle in radians.",
      "format": "number"
    },
    "exact": {
      "label": "In terms of π",
      "description": "The reference angle as an exact multiple of π.",
      "format": "text"
    },
    "coterminal": {
      "label": "Coterminal angle (°)",
      "description": "The angle from 0° to under 360° that ends in the same place.",
      "format": "number"
    },
    "quadrant": {
      "label": "Quadrant",
      "description": "Where the angle’s terminal side lies.",
      "format": "text"
    },
    "x": {
      "label": "Point x (cos θ)",
      "description": "The x value of the angle’s point on the unit circle.",
      "format": "number"
    },
    "y": {
      "label": "Point y (sin θ)",
      "description": "The y value of the angle’s point on the unit circle.",
      "format": "number"
    },
    "refX": {
      "label": "Reference point x",
      "description": "cos of the reference angle: the point mirrored into quadrant I.",
      "format": "number"
    },
    "refY": {
      "label": "Reference point y",
      "description": "sin of the reference angle.",
      "format": "number"
    },
    "steps": {
      "label": "Working",
      "description": "How the reference angle is found.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "u": "deg",
      "a": 225,
      "k": "5/3"
    },
    "outputs": {
      "reference": 45,
      "radians": 0.7853981633974483,
      "exact": "π/4",
      "coterminal": 225,
      "quadrant": "Quadrant III",
      "x": -0.7071067811865477,
      "y": -0.7071067811865475,
      "refX": 0.7071067811865476,
      "refY": 0.7071067811865475,
      "steps": "225° is coterminal with 225°; Quadrant III; reference angle = angle − 180° = 45°"
    },
    "text": "The reference angle is 45°."
  },
  "examples": [
    {
      "given": {
        "u": "deg",
        "a": 225
      },
      "expect": {
        "reference": 45,
        "exact": "π/4",
        "coterminal": 225,
        "quadrant": "Quadrant III",
        "radians": 0.7853981633974483
      },
      "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)"
    },
    {
      "given": {
        "u": "pi",
        "k": "5/3"
      },
      "expect": {
        "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)"
    },
    {
      "given": {
        "u": "pi",
        "k": "-1/6"
      },
      "expect": {
        "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); hand calculation in content.mdx: −π/6 + 2π = 11π/6"
    },
    {
      "given": {
        "u": "deg",
        "a": 870
      },
      "expect": {
        "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); hand calculation in content.mdx: 870° − 2 × 360° = 150°, 180° − 150° = 30°"
    },
    {
      "given": {
        "u": "rad",
        "a": 4
      },
      "expect": {
        "radians": 0.8584073464102069,
        "quadrant": "Quadrant III",
        "reference": 49.18311805232929
      },
      "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); hand calculation in content.mdx: 4 − π; Python 3: 4-math.pi, math.degrees(4-math.pi)"
    },
    {
      "given": {
        "u": "deg",
        "a": -90
      },
      "expect": {
        "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); hand calculation in content.mdx: −90° + 360° = 270°, 270° − 180° = 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)"
  ],
  "related": [
    "trigonometry",
    "angle",
    "sine",
    "cosine"
  ],
  "changelog": []
}