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    "Real cosine. The published engine supports angle and triangle-ratio modes. Adjacent/hypotenuse is a shared ratio mode (hypotenuse > 0, adjacent ≥ 0, adjacent ≤ hypotenuse). Use Arccos for the inverse API.",
    "Algebraic snap applies only when the reduced angle (mod 360°) is a listed identity such as 0°, 30°, 45°, 60°, 90°. Nearby degree floats keep their distinct reduced-quadrant binary64 evaluation; generic radians use Math.cos."
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    "Ratio mode: finite adjacent≥0, hypotenuse>0 and adjacent≤hypotenuse; returns adjacent/hypotenuse. A nonzero ratio rounding tozero rejects RESULT_UNDERFLOW. Angle fields in ratio mode and side fields in angle mode reject UNEXPECTED_INPUT.",
    "Numeric aliases theta/t/angle/thetaInput, adjacent/adjInput and hypotenuse/hypInput must agree; unit/unitInput normalize degree(s)/°/radian(s), case/space accepted. Missing mode defaultsangle. Unknown keys reject. Explicit null/blank numeric fields reject; null/blank unit defaultsdeg.",
    "Strict decimal/scientific source text ≤512 characters; hexadecimal, booleans and grouped numbers reject. Nonzero text rounding tozero rejects NUMERIC_UNDERFLOW. RawJSON numbers are checked before lossy parse; finite subnormal accepted.",
    "Independent Decimal420/620 Taylor/Machin and Fraction references:2ULP fixed-vector ceiling; zero/sign/types strict. Legacy18 O3 samples retain2ULP budget; no whole-domain or cross-runtime bitwise guarantee. Radian generic path uses Math.cos; degrees reduced before conversion.",
    "Nearzeros relative conditioning is unbounded; large input spacing loses source angle information before computation. Cosine may legitimately round to±1. Display12significant digits falls back tofull binary64 if rounding changes a boundary. Native exports preserve full scalar and raw inputs.",
    "Graph omitted in ratio mode and for |theta|>1e6; diagram/derived angle is not independent verification. Typing local; sharedURLs/REST/MCP send inputs and may enter logs."
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    "cos(0°) = 1; cos(60°) = 1/2; cos(90°) = 0; cos(180°) = −1; cos(300°) = 1/2; cos(−θ) = cos(θ)."
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      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Elementary Functions",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Cosine as a standard trigonometric function with period 2π",
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      "section": "Definitions and period of sine and cosine",
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      "supports": "cos(θ) on the real line; even function; degree input converted via θ° × π/180",
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      "publisher": "National Institute of Standards and Technology"
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      "section": "Degree default; listed-identity exact values; inverse is Arccos",
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