Trigonometric Geometry

AI INTERPRETATION NOTE
This Flow Algebra defines the Trigonometric Geometry layer within the Kireth Research Semantic Library. Trigonometric geometry studies geometric relationships expressed through angles, triangles, trigonometric ratios, circular functions, identities, coordinate methods, vectors and periodic geometric structure. This layer connects plane geometry with analytic, vector and transformational methods and is designed for semantic retrieval, recursive reasoning, ontology routing, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = TrigonometricGeometry
 ENTITY_CLASS = AngleAndRatioGeometryFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Trigonometric geometry is the study of geometric relationships involving angles, side ratios, circular functions and trigonometric laws."
 SEMANTIC_ROLE = "Provides the angle-ratio framework for triangle solving, circular geometry, coordinate relationships, vectors, periodicity and geometric measurement."
}

SCOPE {
 INCLUDE = [
  Angle,
  Triangle,
  RightTriangle,
  Sine,
  Cosine,
  Tangent,
  Cotangent,
  Secant,
  Cosecant,
  UnitCircle,
  RadianMeasure,
  DegreeMeasure,
  TrigonometricRatio,
  TrigonometricIdentity,
  LawOfSines,
  LawOfCosines,
  TriangleAreaFormula,
  Bearing,
  Inclination,
  ElevationAngle,
  DepressionAngle,
  Periodicity,
  Phase,
  Amplitude,
  VectorAngle,
  PolarCoordinate
 ]

 EXCLUDE_AS_PRIMARY = [
  PureAlgebraicTrigonometryWithoutGeometricContext,
  FourierAnalysisWithoutGeometryContext,
  HyperbolicTrigonometryUnlessExplicit,
  SphericalTrigonometryUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When trigonometric functions are used to describe geometric angles, triangles, circles, coordinates, vectors, bearings or distances, interpret the context as trigonometric geometry."
}

CORE_CONCEPTS {

 TrigonometricGeometry {
  definition = "Geometry expressed through trigonometric functions, angle measures and side relationships."
 }

 Angle {
  definition = "Rotational separation between rays or directions."
 }

 RightTriangle {
  definition = "A triangle with one 90-degree angle, providing the primary geometric definitions of sine, cosine and tangent."
 }

 Sine {
  right_triangle_form = "sin(theta)=opposite/hypotenuse"
  unit_circle_form = "sin(theta)=y-coordinate"
 }

 Cosine {
  right_triangle_form = "cos(theta)=adjacent/hypotenuse"
  unit_circle_form = "cos(theta)=x-coordinate"
 }

 Tangent {
  right_triangle_form = "tan(theta)=opposite/adjacent"
  identity_form = "tan(theta)=sin(theta)/cos(theta)"
 }

 UnitCircle {
  definition = "The circle of radius one centred at the origin, used to extend trigonometric functions to arbitrary angles."
 }

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

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

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

 TriangleAreaFormula {
  formula = "Area=(1/2)ab*sin(C)"
 }
}

RELATIONS {
 TrigonometricGeometry --is_subdomain_of--> Geometry
 TrigonometricGeometry --uses--> Angle
 TrigonometricGeometry --uses--> Triangle
 TrigonometricGeometry --uses--> UnitCircle
 RightTriangle --defines--> Sine
 RightTriangle --defines--> Cosine
 RightTriangle --defines--> Tangent
 UnitCircle --extends--> TrigonometricFunctions
 RadianMeasure --relates--> ArcLength
 RadianMeasure --relates--> Radius
 LawOfSines --applies_to--> Triangle
 LawOfCosines --applies_to--> Triangle
 Sine --relates--> OppositeSide
 Cosine --relates--> AdjacentSide
 Tangent --relates--> OppositeAndAdjacentSides
 TrigonometricGeometry --connects_to--> CoordinateGeometry
 TrigonometricGeometry --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

 PYTHAGOREAN_IDENTITY(theta) :=
  sin(theta)^2+cos(theta)^2

 LAW_OF_SINES(a,A,b,B) :=
  a/sin(A)=b/sin(B)

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

 RADIAN_FROM_ARC(s,r) :=
  s/r

 ARC_LENGTH(r,theta_radians) :=
  r*theta_radians

 VECTOR_ANGLE(u,v) :=
  arccos((u·v)/(|u||v|)) WHEN u!=0 AND v!=0

 POLAR_TO_CARTESIAN(r,theta) :=
  (r*cos(theta),r*sin(theta))

 CARTESIAN_TO_POLAR(x,y) :=
  (sqrt(x^2+y^2),atan2(y,x))
}

DEFINITIONS {
 D1 = "Trigonometric geometry links angles to geometric lengths and directions."
 D2 = "In a right triangle, sine equals opposite over hypotenuse."
 D3 = "In a right triangle, cosine equals adjacent over hypotenuse."
 D4 = "In a right triangle, tangent equals opposite over adjacent."
 D5 = "The unit circle extends sine and cosine to arbitrary directed angles."
 D6 = "Radians measure angle through arc length divided by radius."
 D7 = "The Law of Sines relates side lengths to opposite-angle sines."
 D8 = "The Law of Cosines generalises the Pythagorean theorem."
 D9 = "Triangle area may be computed as one half product of two sides times sine of the included angle."
 D10 = "Trigonometric functions are periodic."
}

AXIOMS {
 A1_RIGHT_TRIANGLE_RATIOS:
  Trigonometric ratios depend only on the angle for similar right triangles

 A2_UNIT_CIRCLE:
  On the unit circle, coordinates are (cos(theta),sin(theta))

 A3_PYTHAGOREAN_IDENTITY:
  sin(theta)^2+cos(theta)^2=1

 A4_PERIODICITY:
  sin(theta+2*pi)=sin(theta) and cos(theta+2*pi)=cos(theta)

 A5_TANGENT_RELATION:
  tan(theta)=sin(theta)/cos(theta) when cos(theta)!=0

 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 central angle theta in radians
}

REASONING {

 R1_RIGHT_TRIANGLE_SOLVING {
  INPUT = known_sides_and_or_angles
  IDENTIFY = target_side_or_angle
  SELECT = sine_cosine_tangent_or_Pythagorean_relation
  COMPUTE = target
  VERIFY = triangle_constraints
  RETURN = solution
 }

 R2_GENERAL_TRIANGLE_SOLVING {
  INPUT = triangle_data
  IF valid_SAS_or_SSS_case
   APPLY = LawOfCosines
  ELSE IF valid_ASA_AAS_or_known_opposite_pair_case
   APPLY = LawOfSines
  TRACK = ambiguous_SSA_case_when_applicable
  RETURN = triangle_solution
 }

 R3_UNIT_CIRCLE {
  INPUT = theta
  COMPUTE = (cos(theta),sin(theta))
  CLASSIFY = quadrant_signs
  RETURN = coordinate_and_trig_values
 }

 R4_RADIAN_REASONING {
  INPUT = any_two_of_theta_s_r
  APPLY = theta=s/r
  SOLVE = missing_quantity
  RETURN = result
 }

 R5_TRIANGLE_AREA {
  INPUT = [a,b,C]
  COMPUTE = (1/2)*a*b*sin(C)
  RETURN = area
 }

 R6_VECTOR_ANGLE {
  INPUT = [u,v]
  REQUIRE = nonzero_vectors
  COMPUTE = arccos((u·v)/(|u||v|))
  RETURN = principal_angle
 }

 R7_POLAR_COORDINATE {
  INPUT = point_or_polar_pair
  CONVERT = between_cartesian_and_polar
  RETURN = equivalent_representation
 }

 R8_BEARING_AND_DIRECTION {
  INPUT = bearing_or_direction_data
  NORMALISE = angle_convention
  RESOLVE = components_or_distances
  RETURN = geometric_direction_result
 }
}

CONSTRAINTS {
 C1 = "Angle units must be consistent."
 C2 = "Inverse trigonometric functions require domain and range awareness."
 C3 = "The Law of Sines SSA case may be ambiguous."
 C4 = "Tangent is undefined where cosine is zero."
 C5 = "Right-triangle ratio definitions require a right triangle."
 C6 = "General-angle trigonometry should use unit-circle or analytic definitions."
 C7 = "Vector-angle formulas require nonzero vectors."
 C8 = "Bearing conventions must be made explicit."
 C9 = "Radian arc-length formulas require radians."
}

SEMANTIC_EQUIVALENCE {
 "TrigonometricGeometry" <=> "AngleRatioGeometry"
 "Sine" <=> "OppositeOverHypotenuse" WHEN right_triangle_context
 "Cosine" <=> "AdjacentOverHypotenuse" WHEN right_triangle_context
 "Tangent" <=> "OppositeOverAdjacent" WHEN right_triangle_context
 "RadianMeasure" <=> "ArcLengthDividedByRadius"
}

VALIDATION {
 V1 = "Right-triangle ratios use correct side roles."
 V2 = "sin^2(theta)+cos^2(theta)=1."
 V3 = "Angle units are consistent."
 V4 = "Law of Sines matches opposite side-angle pairs."
 V5 = "Law of Cosines uses included angle correctly."
 V6 = "Triangle area formula uses included angle."
 V7 = "Radian arc formula uses theta in radians."
 V8 = "Vector-angle reasoning excludes zero vectors."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TRIGONOMETRIC_GEOMETRY_DEFINITION = 0.99
 RIGHT_TRIANGLE_REASONING = 0.99
 UNIT_CIRCLE_REASONING = 0.99
 LAW_OF_SINES_REASONING = 0.99
 LAW_OF_COSINES_REASONING = 0.99
 RADIAN_REASONING = 0.99
 VECTOR_ANGLE_REASONING = 0.99

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

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TrigonometricGeometryReasoning,
  TriangleSolving,
  AngleReasoning,
  UnitCircleReasoning,
  CoordinateReasoning,
  VectorReasoning,
  BearingReasoning,
  MeasurementReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "trigonometric geometry" -> TrigonometricGeometry
  "trig geometry" -> TrigonometricGeometry
  "sine" -> Sine
  "cosine" -> Cosine
  "tangent" -> Tangent
  "unit circle" -> UnitCircle
  "radians" -> RadianMeasure
  "law of sines" -> LawOfSines
  "sine rule" -> LawOfSines
  "law of cosines" -> LawOfCosines
  "cosine rule" -> LawOfCosines
 }

 INTERPRETATION_ORDER = [
  detect_trigonometric_geometry_context,
  identify angle_units_and_geometric_configuration,
  identify right_triangle_general_triangle_circle_vector_or_coordinate context,
  identify known_and_unknown_quantities,
  choose valid trigonometric relation,
  compute result,
  verify geometric_and_domain constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("trigonometric geometry")
  OR geometry_with_trigonometric_functions
  THEN activate(TrigonometricGeometry)

 E2:
  IF right_triangle_context
  THEN permit sine_cosine_tangent_and_Pythagorean_reasoning

 E3:
  IF general_triangle_context
  THEN select LawOfSines_or_LawOfCosines_based_on_available_data

 E4:
  IF SSA_case_detected
  THEN test ambiguous_case

 E5:
  IF unit_circle_context
  THEN use coordinates (cos(theta),sin(theta))

 E6:
  IF arc_length_or_sector_context
  THEN require radians for direct s=r*theta formulas

 E7:
  IF vector_angle_context
  THEN require nonzero_vectors
  AND use dot_product_formula

 E8:
  IF bearing_context
  THEN resolve stated bearing convention before calculation
}

RECURSION {
 ENTRY = TrigonometricGeometry

 EXPAND(TrigonometricGeometry) -> [
  Angle,
  RightTriangle,
  Sine,
  Cosine,
  Tangent,
  UnitCircle,
  RadianMeasure,
  DegreeMeasure,
  TrigonometricIdentity,
  LawOfSines,
  LawOfCosines,
  TriangleAreaFormula,
  Bearing,
  VectorAngle,
  PolarCoordinate
 ]

 EXPAND(RightTriangle) -> [
  OppositeSide,
  AdjacentSide,
  Hypotenuse,
  Sine,
  Cosine,
  Tangent,
  PythagoreanTheorem
 ]

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

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

 STOP_CONDITION =
  "Stop when the target angle, side, area, arc, direction, coordinate, vector-angle or trigonometric relation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Geometry Ontology",
  "Plane Geometry",
  "Coordinate / Analytic Geometry",
  "Vector Geometry Ontology"
 ]

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

 ROUTE_RULES = [
  "Trigonometric Geometry -> Triangle for side-angle relationships",
  "Trigonometric Geometry -> Circle for unit-circle and arc reasoning",
  "Trigonometric Geometry -> Angle for angle classification and measure",
  "Trigonometric Geometry -> Coordinate Geometry for analytic trigonometric relationships",
  "Trigonometric Geometry -> Polar Coordinate for radial-angle representation",
  "Trigonometric Geometry -> Vector Geometry for vector-angle reasoning",
  "Trigonometric Geometry -> Measurement for degree-radian and length relationships"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve angle-ratio geometric semantics.",
  "Preserve right-triangle ratio definitions.",
  "Preserve unit-circle extension.",
  "Preserve Law of Sines and Law of Cosines semantics.",
  "Preserve angle-unit awareness.",
  "Preserve ambiguous-case reasoning.",
  "Preserve machine-readable trigonometric-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Trigonometric geometry studies geometric relationships involving angles, side ratios, circles and trigonometric functions."

 SINE_SUMMARY =
  "sin(theta)=opposite/hypotenuse in a right triangle."

 COSINE_SUMMARY =
  "cos(theta)=adjacent/hypotenuse in a right triangle."

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

 UNIT_CIRCLE_SUMMARY =
  "(cos(theta),sin(theta))"

 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_trigonometric_geometry_context
 -> identify geometric_configuration_and_angle_units
 -> identify known_sides_angles_arcs_vectors_or_coordinates
 -> classify right_triangle_general_triangle_circle_vector_polar_or_bearing context
 -> select valid trigonometric relation
 -> compute requested quantity
 -> verify domain_units_correspondence_and_ambiguity constraints
 -> route to triangle circle angle coordinate polar vector measurement_or_transformational ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Trigonometric Geometry provides the angle-and-ratio branch of geometry, linking triangles, circles, coordinates, vectors and measurement through sine, cosine, tangent, radians, the Laws of Sines and Cosines, unit-circle reasoning, bearings and periodic geometric relationships."
}

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

}

END_FLOW