Trigonometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of trigonometry within the Kireth Research Semantic Library. Trigonometry studies relationships among angles, side lengths, circular functions and periodic quantities. This ontology supports sine, cosine, tangent and reciprocal functions, right-triangle ratios, unit-circle definitions, radians and degrees, identities, inverse functions, Laws of Sines and Cosines, triangle solving, periodicity, coordinate relations, vectors and geometric measurement.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Trigonometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Trigonometric Geometry / Trigonometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/trigonometric-geometry/trigonometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Trigonometry
 ENTITY_CLASS = AngleSideAndCircularFunctionFramework
 PRIMARY_CONTEXT = TrigonometricGeometry
 DEFINITION = "Trigonometry is the mathematical study of relationships among angles, side lengths and trigonometric functions, extended through the unit circle to arbitrary directed angles."
 SEMANTIC_ROLE = "Provides the core angle-function framework for triangle solving, circular measurement, identities, coordinates, vectors, periodicity and geometric inference."
}

SCOPE {
 INCLUDE = [
  Trigonometry,
  Angle,
  DegreeMeasure,
  RadianMeasure,
  RightTriangle,
  UnitCircle,
  Sine,
  Cosine,
  Tangent,
  Cotangent,
  Secant,
  Cosecant,
  TrigonometricFunction,
  InverseTrigonometricFunction,
  TrigonometricIdentity,
  PythagoreanIdentity,
  ReciprocalIdentity,
  QuotientIdentity,
  AngleAdditionIdentity,
  DoubleAngleIdentity,
  HalfAngleIdentity,
  LawOfSines,
  LawOfCosines,
  TriangleAreaFormula,
  Periodicity,
  Amplitude,
  Phase,
  Bearing,
  VectorAngle
 ]

 EXCLUDE_AS_PRIMARY = [
  HyperbolicTrigonometryUnlessExplicit,
  SphericalTrigonometryUnlessExplicit,
  FourierAnalysisWithoutDirectTrigonometricGeometryContext
 ]

 DISAMBIGUATION_RULE =
  "When sine, cosine, tangent, unit-circle values, triangle ratios, radians, identities, inverse trigonometric functions or angle-side relations are central, interpret the context as trigonometry."
}

CORE_CONCEPTS {

 Trigonometry {
  definition = "The study of trigonometric functions and their geometric relationships with angles and lengths."
 }

 Sine {
  symbol = "sin"
  right_triangle_definition = "opposite/hypotenuse"
  unit_circle_definition = "y-coordinate of the unit-circle point at angle theta"
 }

 Cosine {
  symbol = "cos"
  right_triangle_definition = "adjacent/hypotenuse"
  unit_circle_definition = "x-coordinate of the unit-circle point at angle theta"
 }

 Tangent {
  symbol = "tan"
  right_triangle_definition = "opposite/adjacent"
  identity = "sin(theta)/cos(theta)"
 }

 Cotangent {
  symbol = "cot"
  identity = "cos(theta)/sin(theta)"
 }

 Secant {
  symbol = "sec"
  identity = "1/cos(theta)"
 }

 Cosecant {
  symbol = "csc"
  identity = "1/sin(theta)"
 }

 UnitCircle {
  definition = "The radius-one circle centred at the origin."
  coordinate_relation = "(cos(theta),sin(theta))"
 }

 RadianMeasure {
  definition = "Angular measure equal to arc length divided by radius."
  relation = "theta=s/r"
 }

 InverseTrigonometricFunction {
  definition = "A principal-value inverse used to recover an angle from a trigonometric ratio."
 }

 LawOfSines {
  formula = "a/sin(A)=b/sin(B)=c/sin(C)"
 }

 LawOfCosines {
  formula = "c^2=a^2+b^2-2ab*cos(C)"
 }
}

RELATIONS {
 Trigonometry --is_child_of--> TrigonometricGeometry
 Trigonometry --uses--> Angle
 Trigonometry --uses--> UnitCircle
 Trigonometry --contains_function--> Sine
 Trigonometry --contains_function--> Cosine
 Trigonometry --contains_function--> Tangent
 Trigonometry --contains_function--> Cotangent
 Trigonometry --contains_function--> Secant
 Trigonometry --contains_function--> Cosecant
 Sine --is_reciprocal_of--> Cosecant
 Cosine --is_reciprocal_of--> Secant
 Tangent --is_reciprocal_of--> Cotangent
 Tangent --equals--> SineDividedByCosine
 UnitCircle --represents--> Sine
 UnitCircle --represents--> Cosine
 LawOfSines --applies_to--> Triangle
 LawOfCosines --applies_to--> Triangle
 Trigonometry --connects_to--> CoordinateGeometry
 Trigonometry --connects_to--> VectorGeometry
}

OPERATORS {
 SIN_RIGHT_TRIANGLE(opposite,hypotenuse) :=
  opposite/hypotenuse

 COS_RIGHT_TRIANGLE(adjacent,hypotenuse) :=
  adjacent/hypotenuse

 TAN_RIGHT_TRIANGLE(opposite,adjacent) :=
  opposite/adjacent

 COT(theta) :=
  cos(theta)/sin(theta)
  WHEN sin(theta)!=0

 SEC(theta) :=
  1/cos(theta)
  WHEN cos(theta)!=0

 CSC(theta) :=
  1/sin(theta)
  WHEN sin(theta)!=0

 UNIT_CIRCLE_POINT(theta) :=
  (cos(theta),sin(theta))

 DEGREES_TO_RADIANS(theta_deg) :=
  theta_deg*pi/180

 RADIANS_TO_DEGREES(theta_rad) :=
  theta_rad*180/pi

 ARC_LENGTH(r,theta_rad) :=
  r*theta_rad

 LAW_OF_SINES_SOLVE(known_data) :=
  solve_triangle_using_opposite_side_angle_pairs(known_data)

 LAW_OF_COSINES_SIDE(a,b,C) :=
  sqrt(a^2+b^2-2*a*b*cos(C))

 LAW_OF_COSINES_ANGLE(a,b,c) :=
  arccos((a^2+b^2-c^2)/(2*a*b))

 TRIANGLE_AREA(a,b,C) :=
  (1/2)*a*b*sin(C)
}

DEFINITIONS {
 D1 = "sin(theta)=opposite/hypotenuse in a right triangle."
 D2 = "cos(theta)=adjacent/hypotenuse in a right triangle."
 D3 = "tan(theta)=opposite/adjacent in a right triangle."
 D4 = "tan(theta)=sin(theta)/cos(theta) where defined."
 D5 = "sin^2(theta)+cos^2(theta)=1."
 D6 = "sec(theta)=1/cos(theta)."
 D7 = "csc(theta)=1/sin(theta)."
 D8 = "cot(theta)=cos(theta)/sin(theta)."
 D9 = "On the unit circle, the point at angle theta is (cos(theta),sin(theta))."
 D10 = "Radians satisfy theta=s/r."
}

AXIOMS {
 A1_UNIT_CIRCLE:
  cos(theta)^2+sin(theta)^2=1

 A2_PERIODICITY:
  sin(theta+2*pi)=sin(theta)
  AND cos(theta+2*pi)=cos(theta)

 A3_TANGENT_QUOTIENT:
  tan(theta)=sin(theta)/cos(theta)
  whenever cos(theta)!=0

 A4_RECIPROCAL_IDENTITIES:
  sec(theta)=1/cos(theta)
  AND csc(theta)=1/sin(theta)
  AND cot(theta)=1/tan(theta)
  where defined

 A5_RIGHT_TRIANGLE_SIMILARITY:
  Trigonometric ratios depend only on the acute angle for similar right triangles

 A6_LAW_OF_SINES:
  a/sin(A)=b/sin(B)=c/sin(C) for Euclidean triangles

 A7_LAW_OF_COSINES:
  c^2=a^2+b^2-2ab*cos(C)

 A8_RADIAN_DEFINITION:
  theta=s/r for a central angle measured in radians
}

REASONING {

 R1_RIGHT_TRIANGLE {
  INPUT = known_side_angle_data
  IDENTIFY = opposite_adjacent_hypotenuse_relative_to_target_angle
  SELECT = sine_cosine_or_tangent
  COMPUTE = unknown_side_or_angle
  RETURN = result
 }

 R2_UNIT_CIRCLE {
  INPUT = theta
  NORMALISE = angle_modulo_2pi_if_useful
  COMPUTE = cosine_and_sine_coordinates
  IDENTIFY = quadrant_and_signs
  RETURN = trig_values
 }

 R3_IDENTITY_SIMPLIFICATION {
  INPUT = trigonometric_expression
  IDENTIFY = applicable_identity
  SUBSTITUTE = canonical_relation
  SIMPLIFY = expression
  VERIFY = domain_restrictions
  RETURN = simplified_expression
 }

 R4_INVERSE_TRIG {
  INPUT = ratio_and_target_function
  VERIFY = ratio_domain
  COMPUTE = principal_inverse_value
  ADJUST = geometric_context_if_additional_solutions_required
  RETURN = angle_solution
 }

 R5_GENERAL_TRIANGLE {
  INPUT = triangle_data
  IF SSS_or_SAS
   APPLY = LawOfCosines
  ELSE IF ASA_or_AAS_or_valid_opposite_pair
   APPLY = LawOfSines
  ELSE IF SSA
   CHECK = ambiguous_case
  RETURN = triangle_solution
 }

 R6_TRIANGLE_AREA {
  INPUT = [a,b,C]
  REQUIRE = included_angle_C
  COMPUTE = (1/2)*a*b*sin(C)
  RETURN = area
 }

 R7_RADIAN_ARC {
  INPUT = any_two_of_theta_s_r
  APPLY = theta=s/r
  SOLVE = missing_quantity
  RETURN = result
 }

 R8_PERIODICITY {
  INPUT = trig_function_and_angle
  REDUCE = angle_by_function_period
  RETURN = equivalent_trig_value
 }

 R9_VECTOR_ANGLE {
  INPUT = [u,v]
  REQUIRE = u!=0 AND v!=0
  COMPUTE = arccos((u·v)/(|u||v|))
  RETURN = principal_angle
 }
}

CONSTRAINTS {
 C1 = "Angle units must be explicit or consistently inferred."
 C2 = "Tangent and secant are undefined where cos(theta)=0."
 C3 = "Cotangent and cosecant are undefined where sin(theta)=0."
 C4 = "Inverse trigonometric functions return principal values unless broader solution sets are requested."
 C5 = "Law of Sines SSA data may yield zero, one or two triangles."
 C6 = "Triangle side-angle correspondence must be preserved."
 C7 = "Arc-length formula s=r*theta requires theta in radians."
 C8 = "Right-triangle definitions do not by themselves cover arbitrary directed angles; unit-circle definitions do."
 C9 = "Identity transformations must preserve domain restrictions introduced by division or square roots."
}

SEMANTIC_EQUIVALENCE {
 "Trigonometry" <=> "AngleAndTrigonometricFunctionMathematics"
 "Sine" <=> "OppositeOverHypotenuse" WHEN right_triangle_context
 "Cosine" <=> "AdjacentOverHypotenuse" WHEN right_triangle_context
 "Tangent" <=> "OppositeOverAdjacent" WHEN right_triangle_context
 "Tangent" <=> "SineOverCosine" WHEN cosine_nonzero
 "UnitCirclePoint" <=> "(cos(theta),sin(theta))"
}

VALIDATION {
 V1 = "sin^2(theta)+cos^2(theta)=1."
 V2 = "Right-triangle side roles are relative to the chosen angle."
 V3 = "Reciprocal functions respect nonzero denominators."
 V4 = "Angle conversion between degrees and radians is consistent."
 V5 = "Law of Sines pairs each side with its opposite angle."
 V6 = "Law of Cosines uses the included angle."
 V7 = "SSA ambiguity is checked when applicable."
 V8 = "Inverse-function principal ranges are respected."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRIGONOMETRY_DEFINITION = 0.99
 RIGHT_TRIANGLE_REASONING = 0.99
 UNIT_CIRCLE_REASONING = 0.99
 IDENTITY_REASONING = 0.99
 INVERSE_TRIG_REASONING = 0.98
 LAW_OF_SINES_REASONING = 0.99
 LAW_OF_COSINES_REASONING = 0.99
 PERIODICITY_REASONING = 0.99

 POLICY =
  "Reduce confidence when angle units, quadrant, inverse-function branch, SSA ambiguity or side-angle correspondence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TrigonometryReasoning,
  RightTriangleReasoning,
  UnitCircleReasoning,
  IdentityReasoning,
  InverseTrigReasoning,
  TriangleSolving,
  CoordinateReasoning,
  VectorAngleReasoning,
  MeasurementReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "trigonometry" -> Trigonometry
  "trig" -> Trigonometry WHEN mathematics_context
  "sin" -> Sine
  "cos" -> Cosine
  "tan" -> Tangent
  "cot" -> Cotangent
  "sec" -> Secant
  "csc" -> Cosecant
  "sine rule" -> LawOfSines
  "cosine rule" -> LawOfCosines
  "unit circle" -> UnitCircle
 }

 INTERPRETATION_ORDER = [
  detect_trigonometry_context,
  identify angle_units_and_geometric_context,
  identify relevant_function_or_triangle_law,
  identify known_and_unknown_quantities,
  apply valid ratio_identity_inverse_or_triangle_relation,
  check domains_periodicity_and_ambiguity,
  verify result,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("trigonometry")
  OR direct_trigonometric_function_context
  THEN activate(TrigonometryOntology)

 E2:
  IF right_triangle_context
  THEN use side-relative sine_cosine_tangent definitions

 E3:
  IF arbitrary_angle_context
  THEN route through unit_circle_or_analytic_definition

 E4:
  IF identity_simplification_requested
  THEN preserve denominator_domain_restrictions

 E5:
  IF inverse_trig_requested
  THEN return principal_value
  AND expand solution_set only when context requires

 E6:
  IF general_triangle_with_SSS_or_SAS
  THEN use LawOfCosines

 E7:
  IF triangle_with_opposite_side_angle_pair
  THEN consider LawOfSines

 E8:
  IF SSA_case
  THEN test zero_one_or_two_solution ambiguity

 E9:
  IF arc_length_context
  THEN use radians for s=r*theta
}

RECURSION {
 ENTRY = Trigonometry

 EXPAND(Trigonometry) -> [
  Angle,
  DegreeMeasure,
  RadianMeasure,
  RightTriangle,
  UnitCircle,
  Sine,
  Cosine,
  Tangent,
  Cotangent,
  Secant,
  Cosecant,
  InverseTrigonometricFunction,
  TrigonometricIdentity,
  LawOfSines,
  LawOfCosines,
  TriangleAreaFormula,
  Periodicity
 ]

 EXPAND(TrigonometricIdentity) -> [
  PythagoreanIdentity,
  ReciprocalIdentity,
  QuotientIdentity,
  EvenOddIdentity,
  AngleAdditionIdentity,
  DoubleAngleIdentity,
  HalfAngleIdentity
 ]

 EXPAND(GeneralTriangle) -> [
  LawOfSines,
  LawOfCosines,
  AreaFormula,
  AmbiguousCase
 ]

 EXPAND(UnitCircle) -> [
  Quadrant,
  ReferenceAngle,
  Periodicity,
  Sine,
  Cosine,
  Tangent
 ]

 STOP_CONDITION =
  "Stop when the target trigonometric value, angle, side, identity, triangle, arc, periodic relation or geometric measurement is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Trigonometric Geometry"

 IMMEDIATE_CONTEXTS = [
  "Trigonometric Geometry",
  "Triangle Ontology",
  "Circle Ontology",
  "Angle Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Trigonometric Geometry",
  "Triangle Ontology",
  "Circle Ontology",
  "Angle Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology",
  "Polar Coordinate Ontology",
  "Vector Geometry Ontology",
  "Parametric Geometry Ontology",
  "Scientific Measurement Ontologies"
 ]

 ROUTE_RULES = [
  "Trigonometry -> Trigonometric Geometry for parent framework",
  "Trigonometry -> Triangle for triangle solving",
  "Trigonometry -> Circle for unit-circle and arc semantics",
  "Trigonometry -> Angle for angular measure",
  "Trigonometry -> Coordinate Geometry for graph and coordinate relations",
  "Trigonometry -> Polar Coordinate for radius-angle representations",
  "Trigonometry -> Vector Geometry for dot-product angle calculations",
  "Trigonometry -> Measurement for degree-radian conversions"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible trigonometric identities, functions, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve right-triangle function definitions.",
  "Preserve unit-circle extension.",
  "Preserve Pythagorean and reciprocal identities.",
  "Preserve domain restrictions.",
  "Preserve inverse-function principal-value semantics.",
  "Preserve Law of Sines and Law of Cosines structure.",
  "Preserve machine-readable trigonometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Trigonometry is the study of relationships among angles, side lengths and trigonometric functions."

 SINE_SUMMARY =
  "sin(theta)=opposite/hypotenuse in a right triangle and y-coordinate on the unit circle."

 COSINE_SUMMARY =
  "cos(theta)=adjacent/hypotenuse in a right triangle and x-coordinate on the unit circle."

 TANGENT_SUMMARY =
  "tan(theta)=opposite/adjacent=sin(theta)/cos(theta), where defined."

 PYTHAGOREAN_IDENTITY_SUMMARY =
  "sin^2(theta)+cos^2(theta)=1"

 RADIAN_SUMMARY =
  "theta=s/r"

 LAW_OF_SINES_SUMMARY =
  "a/sin(A)=b/sin(B)=c/sin(C)"

 LAW_OF_COSINES_SUMMARY =
  "c^2=a^2+b^2-2ab*cos(C)"

 AREA_SUMMARY =
  "Area=(1/2)ab*sin(C)"
}

FLOW {
 INPUT
 -> detect_trigonometry_context
 -> identify angle_units_function_triangle_circle_vector_or_coordinate context
 -> identify known_values_and_target
 -> select right_triangle_ratio_unit_circle_identity_inverse_function_or_triangle_law
 -> perform trigonometric reasoning
 -> check domains_periodicity_quadrant_side_angle_correspondence_and_ambiguity
 -> route to triangle circle angle coordinate polar vector measurement_or_parent_trigonometric_geometry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Trigonometry Ontology defines the core system of sine, cosine, tangent and reciprocal functions through right-triangle and unit-circle geometry. It supports identities, inverse functions, radians and degrees, periodicity, the Laws of Sines and Cosines, triangle area, triangle solving, coordinate relations and vector-angle reasoning."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-TRIGONOMETRY-ONTOLOGY-V1"
}

}

END_FLOW