{
  "id": "inverse-cosine",
  "version": "c662f292890f",
  "status": "published",
  "name": "Inverse Cosine Calculator",
  "question": "What is the inverse cosine of x?",
  "summary": "Finds arccos x, the angle from 0° to 180° whose cosine is x, in degrees and radians, exact as a multiple of π for the special values.",
  "category": "math",
  "subcategory": "trigonometry",
  "url": "https://www.acalculator.org/math/inverse-cosine-calculator",
  "markdown": "https://www.acalculator.org/math/inverse-cosine-calculator.md",
  "kind": "function",
  "method": "arccos x is the angle θ from 0° to 180° (0 to π) with cos θ = x. When x is within 10⁻¹⁴ of the cosine of a multiple of 15° in that range, the angle is that multiple exactly; otherwise it is computed in 64-bit floating point.",
  "assumptions": [
    "x is from −1 to 1; the cosine of a real angle is never outside that range.",
    "The answer is the principal value. Other angles with the same cosine are −θ and those angles plus whole turns."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "x": {
        "title": "Value x",
        "description": "A number from −1 to 1: the cosine of the angle.",
        "type": "number",
        "minimum": -1,
        "maximum": 1
      }
    }
  },
  "outputs": {
    "angle": {
      "label": "arccos x",
      "description": "The principal angle whose cos is x, in degrees.",
      "format": "quantity",
      "unit": "deg"
    },
    "radians": {
      "label": "In radians",
      "description": "The same angle in radians.",
      "format": "number"
    },
    "exact": {
      "label": "Exact angle",
      "description": "The angle as a multiple of π, when x is the value at a multiple of 15°.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "x": 0.5
    },
    "outputs": {
      "angle": 1.0471975511965976,
      "radians": 1.0471975511965976,
      "exact": "π/3 = 60° (cos 60° = 1/2)"
    },
    "text": "arccos(0.5) = 60°."
  },
  "examples": [
    {
      "given": {
        "x": 0.5
      },
      "expect": {
        "angle": 1.0471975511965976,
        "radians": 1.0471975511965976,
        "exact": "π/3 = 60° (cos 60° = 1/2)"
      },
      "source": "hand calculation in content.mdx: cos 60° = 1/2 and 60° is in [0°, 180°]; OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions"
    },
    {
      "given": {
        "x": -1
      },
      "expect": {
        "angle": 3.141592653589793,
        "exact": "π = 180° (cos 180° = -1)"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions"
    },
    {
      "given": {
        "x": -0.8660254037844386
      },
      "expect": {
        "angle": 2.6179938779914944,
        "exact": "5π/6 = 150° (cos 150° = -√3/2)"
      },
      "source": "hand calculation in content.mdx: cos 150° = −cos 30° = −√3/2; OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions"
    },
    {
      "given": {
        "x": 0.3
      },
      "expect": {
        "angle": 1.266103672779499
      },
      "source": "NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23; Python 3: math.acos(0.3)"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions (retrieved 2026-10-01)",
    "OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-01)",
    "NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (principal values). https://dlmf.nist.gov/4.23 (retrieved 2026-10-01)"
  ],
  "related": [
    "cosine",
    "inverse-sine",
    "inverse-tangent",
    "law-of-cosines"
  ],
  "changelog": []
}