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  "api": "https://www.calculatorx.com/api/v1/calc/sin",
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    "Real sine. The published engine is forward sin(θ). Opposite/hypotenuse is a local triangle convenience (hypotenuse > 0, opposite ≥ 0, opposite ≤ hypotenuse). Use Arcsin 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 floats, including 30.00000005°, keep IEEE-754 Math.sin."
  ],
  "limitations": [
    "Angle mode: finite binary64 theta; a nonzero degree sine rounding tozero rejects RESULT_UNDERFLOW; deg(default) or rad. Degree reduction uses signed remainder before quadrant reduction; no add360 or angle-proximity epsilon. Listed common degree angles and 17 normalized binary64 radian tokens within ±2π use symbolic recognition, not mathematical exactness of floating-point radians.",
    "Ratio mode: finite opposite≥0, hypotenuse>0 and opposite≤hypotenuse; returns opposite/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, opposite/oppInput 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.sin; degrees reduced before conversion.",
    "Nearzeros relative conditioning is unbounded; large input spacing loses source angle information before computation. Sine 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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    "Missing θ returns MISSING_REQUIRED_INPUT. Invalid unit returns INVALID_UNIT. Malformed numbers, NaN, and ±Infinity return INVALID_NUMBER.",
    "sin(0°) = 0; sin(30°) = 1/2; sin(90°) = 1; sin(150°) = 1/2; sin(180°) = 0; sin(−θ) = −sin(θ)."
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    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Elementary Functions",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Sine as a standard trigonometric function with period 2π",
      "kind": "reference"
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      "title": "NIST DLMF §4.14",
      "section": "Definitions and period of sine and cosine",
      "url": "https://dlmf.nist.gov/4.14",
      "supports": "sin(θ) on the real line; odd function; degree input converted via θ° × π/180",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
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    {
      "title": "CalculatorX mathematical conventions",
      "section": "Degree default; listed-identity exact values; inverse is Arcsin",
      "supports": "Published engine is forward sin(θ) only. Algebraic snap is identity-only; |y|<1e-12 display rounding is UI-only.",
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