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  "page": "https://www.calculatorx.com/calc/math/arctan",
  "api": "https://www.calculatorx.com/api/v1/calc/arctan",
  "schema_input": "https://www.calculatorx.com/schemas/math.arctan.input.json",
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  "assumptions": [
    "Real inverse tangent with principal branch (−π/2, π/2) (NIST DLMF §4.23(ii)). Infinity is not accepted as an input.",
    "Opposite/adjacent in the UI is a right-triangle length convenience (opposite ≥ 0, adjacent > 0, same unit). It is not atan2. The published engine remains single-argument arctan(x). Use Decimal for a signed ratio, and Tan for the forward map.",
    "When x is exactly a listed common value (0, ±1/√3, ±1, ±√3), the engine returns the exact algebraic angle such as 45° = π/4. Nearby floats are not snapped."
  ],
  "limitations": [
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    "Non-finite → INVALID_NUMBER",
    "Near-common-value hint is display metadata; it does not replace the canonical deg/rad or assert mathematical exactness."
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  "boundary_conditions": [
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    "Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.",
    "arctan(0) = 0°; arctan(1) = 45°; arctan(√3) = 60°; arctan(−1) = −45°.",
    "Large finite |x| approaches ±90°. The mathematical principal range is the open interval (−90°, 90°). IEEE-754 binary64 results may round to ±90° for extremely large finite inputs (for example 1e20 and Number.MAX_VALUE). When the result rounds to ±90°, tan(θ) cannot recover x and verification.passed may be false."
  ],
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      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Elementary Functions",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Inverse trigonometric functions as elementary inverses of sine, cosine, and tangent",
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      "section": "Inverse trigonometric functions; principal values",
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      "supports": "Principal arctan on the real line with range (−π/2, π/2); degree output is θ × 180/π",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
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      "title": "CalculatorX mathematical conventions",
      "section": "Principal branch; exact-value recognition; ratio mode is UI-only",
      "supports": "Published engine is arctan(x) of one real argument (not atan2).",
      "kind": "definition"
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