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  "page": "https://www.calculatorx.com/calc/math/tan",
  "api": "https://www.calculatorx.com/api/v1/calc/tan",
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  "schema_output": "https://www.calculatorx.com/schemas/math.tan.output.json",
  "assumptions": [
    "Real tangent only. Inverse is Arctan, not this published engine.",
    "Documented pole classification: UNDEFINED when distance to 90° + 180°k is < 1e-12°. That threshold is a numerical classification policy, not a claim that those inputs are mathematically exact odd multiples of 90°. |cos(θ)| is not used as a pole test.",
    "If the input is not classified as a pole and distance-to-pole < 0.01°, the engine returns a finite tan(θ) with warnings: [{ code: NEAR_SINGULARITY }]. status remains ok.",
    "Documented common-angle snap: after reducing mod 360°, an input within 1e-12° of a listed common angle (0, ±30, ±45, ±60, …) is replaced by the algebraic value (0, ±1/√3, ±1, ±√3). This is a snap policy, not mathematical exactness. Nearby inputs such as 60.00000005° keep IEEE-754 Math.tan.",
    "Documented cleanup: if no snap applies and |y| < 1e-12, the engine rewrites y to 0. That changes the IEEE-754 result for some non-zero inputs; it is a cleanup policy, not a mathematical identity."
  ],
  "limitations": [
    "Exact poles (90° + 180°k) → UNDEFINED",
    "Within 0.01° of a pole → finite value with warnings NEAR_SINGULARITY",
    "Missing theta → MISSING_REQUIRED_INPUT"
  ],
  "boundary_conditions": [
    "Missing θ returns MISSING_REQUIRED_INPUT. Invalid unit returns INVALID_UNIT. Malformed numbers return INVALID_NUMBER.",
    "tan(0°) = 0; tan(45°) = 1; tan(30°) = 1/√3; tan(60°) = √3; tan(−45°) = −1; tan(180°) = 0; tan(225°) = 1; tan(360°) = 0.",
    "tan(90°), tan(270°), tan(−90°), tan(π/2 rad) return UNDEFINED.",
    "tan(89°) and tan(91°) are large finite values, not UNDEFINED.",
    "Periodicity: tan(θ + 180°) = tan(θ), e.g. tan(405°) = tan(45°) = 1."
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Elementary Functions",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Tangent as sin/cos with period π and poles where cos = 0",
      "kind": "reference"
    },
    {
      "title": "NIST DLMF §4.14",
      "section": "Trigonometric functions",
      "url": "https://dlmf.nist.gov/4.14",
      "supports": "tan(θ) undefined at odd multiples of π/2; degree input converted via θ° × π/180",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
    {
      "title": "CalculatorX mathematical conventions",
      "section": "Poles are UNDEFINED, not Infinity; inverse is Arctan",
      "supports": "Published engine is forward tan(θ) only. Pole classification, common-angle snap, and |y| cleanup are documented numerical policies, not mathematical exactness. Near-pole values stay finite with NEAR_SINGULARITY.",
      "kind": "definition"
    }
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