Sine Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of sine within the Kireth Research Semantic Library. Sine is a trigonometric function relating an angle to a right-triangle side ratio and, more generally, to the vertical coordinate of a point on the unit circle. This ontology supports right-triangle geometry, unit-circle representation, periodicity, symmetry, inverse sine, identities, Laws of Sines, coordinate graphs, vector and wave relationships, domain and range reasoning, and links to cosine, tangent, circles and triangle geometry.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = Sine
 SYMBOLIC_FORMS = [
  "sin(theta)",
  "sin(x)",
  "y=sin(x)"
 ]
 ENTITY_CLASS = TrigonometricFunction
 PRIMARY_CONTEXT = Trigonometry
 DEFINITION = "Sine is the trigonometric function that, in a right triangle, equals opposite side divided by hypotenuse and, on the unit circle, equals the y-coordinate of the point at angle theta."
 SEMANTIC_ROLE = "Provides the vertical circular-coordinate and opposite-to-hypotenuse ratio function for angle, triangle, periodic, identity and geometric measurement reasoning."
}

SCOPE {
 INCLUDE = [
  Sine,
  RightTriangleSine,
  UnitCircleSine,
  SineFunction,
  SineGraph,
  InverseSine,
  Arcsine,
  OppositeSide,
  Hypotenuse,
  Angle,
  UnitCircle,
  Periodicity,
  Amplitude,
  Phase,
  PythagoreanIdentity,
  LawOfSines,
  SineAreaFormula,
  OddFunctionSymmetry,
  ReferenceAngle
 ]

 EXCLUDE_AS_PRIMARY = [
  HyperbolicSineUnlessExplicit,
  SincFunction,
  GeneralWaveformWithoutSineStructure,
  SineOfComplexNumberUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When sine occurs in right-triangle, unit-circle, trigonometric, periodic, Law-of-Sines or geometric-angle contexts, interpret it as the trigonometric sine function."
}

CORE_CONCEPTS {

 Sine {
  symbol = "sin"
  definition = "A trigonometric function assigning each angle theta a value sin(theta)."
  domain_real = "R"
  range_real = "[-1,1]"
 }

 RightTriangleSine {
  definition = "For an acute angle theta in a right triangle, sin(theta)=opposite/hypotenuse."
 }

 UnitCircleSine {
  definition = "For angle theta, sin(theta) is the y-coordinate of the corresponding point on the unit circle."
 }

 SineGraph {
  definition = "The graph y=sin(x), a periodic wave with amplitude 1 and period 2*pi."
 }

 InverseSine {
  symbol = "arcsin"
  alternative_symbol = "sin^-1"
  principal_domain = "[-1,1]"
  principal_range = "[-pi/2,pi/2]"
 }

 Periodicity {
  relation = "sin(theta+2*pi)=sin(theta)"
 }

 OddFunctionSymmetry {
  relation = "sin(-theta)=-sin(theta)"
 }

 PythagoreanIdentity {
  relation = "sin^2(theta)+cos^2(theta)=1"
 }

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

 SineAreaFormula {
  relation = "Area=(1/2)ab*sin(C)"
 }
}

RELATIONS {
 Sine --is_special_case_of--> TrigonometricFunction
 Sine --is_defined_by--> RightTriangleRatio
 Sine --is_defined_by--> UnitCircleCoordinate
 Sine --relates--> OppositeSide
 Sine --relates--> Hypotenuse
 Sine --pairs_with--> Cosine
 Sine --forms_ratio_for--> Tangent
 Sine --has_inverse--> InverseSine
 Sine --has_property--> Periodicity
 Sine --has_property--> OddFunctionSymmetry
 Sine --participates_in--> PythagoreanIdentity
 Sine --participates_in--> LawOfSines
 Sine --participates_in--> TriangleAreaFormula
 UnitCircle --provides--> Sine
}

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

 SIN_UNIT_CIRCLE(theta) :=
  y_coordinate_of_unit_circle_point(theta)

 SIN_FROM_COS(theta,cos_value,sign_context) :=
  sign_context*sqrt(1-cos_value^2)

 ARCSIN(y) :=
  principal_angle_theta_in_[-pi/2,pi/2]
  satisfying sin(theta)=y

 PERIOD_REDUCE(theta) :=
  theta mod 2*pi

 SINE_REFERENCE_ANGLE(theta) :=
  derive_reference_angle_and_sign(theta)

 LAW_OF_SINES_SIDE(b,B,A) :=
  b*sin(A)/sin(B)

 LAW_OF_SINES_ANGLE(a,b,B) :=
  arcsin(a*sin(B)/b)
  subject_to_ambiguous_case

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

DEFINITIONS {
 D1 = "In a right triangle, sin(theta)=opposite/hypotenuse."
 D2 = "On the unit circle, sin(theta) is the y-coordinate."
 D3 = "The real sine function has range [-1,1]."
 D4 = "Sine has period 2*pi."
 D5 = "Sine is an odd function."
 D6 = "sin^2(theta)+cos^2(theta)=1."
 D7 = "sin(theta)=0 at integer multiples of pi."
 D8 = "sin(theta)=1 at theta=pi/2+2k*pi."
 D9 = "sin(theta)=-1 at theta=3pi/2+2k*pi."
 D10 = "arcsin returns principal values in [-pi/2,pi/2]."
}

AXIOMS {
 A1_UNIT_CIRCLE:
  For unit-circle point P(theta)=(cos(theta),sin(theta)), the second coordinate equals sin(theta)

 A2_RIGHT_TRIANGLE:
  In a right triangle sin(theta)=opposite/hypotenuse

 A3_PERIODICITY:
  sin(theta+2*pi)=sin(theta)

 A4_ODD_SYMMETRY:
  sin(-theta)=-sin(theta)

 A5_PYTHAGOREAN_IDENTITY:
  sin^2(theta)+cos^2(theta)=1

 A6_RANGE:
  -1<=sin(theta)<=1 for real theta

 A7_ZERO_SET:
  sin(theta)=0 iff theta=k*pi for integer k

 A8_INVERSE_PRINCIPAL:
  arcsin maps [-1,1] to [-pi/2,pi/2]
}

REASONING {

 R1_RIGHT_TRIANGLE_VALUE {
  INPUT = [opposite,hypotenuse]
  REQUIRE = hypotenuse>0
  COMPUTE = opposite/hypotenuse
  VERIFY = ratio_in_[0,1] for acute_triangle_context
  RETURN = sine_value
 }

 R2_ANGLE_FROM_RATIO {
  INPUT = y
  REQUIRE = -1<=y<=1
  COMPUTE = arcsin(y)
  RETURN = principal_angle
 }

 R3_UNIT_CIRCLE_VALUE {
  INPUT = theta
  REDUCE = modulo_2pi_if_useful
  IDENTIFY = quadrant
  COMPUTE = y_coordinate
  RETURN = sin(theta)
 }

 R4_REFERENCE_ANGLE {
  INPUT = theta
  DETERMINE = reference_angle_alpha
  DETERMINE = sign_by_quadrant
  RETURN = sign*sin(alpha)
 }

 R5_IDENTITY {
  INPUT = cosine_or_expression_data
  APPLY = sin^2(theta)=1-cos^2(theta)
  RESOLVE = sign_from_context_when_square_root_taken
  RETURN = sine_or_simplified_expression
 }

 R6_LAW_OF_SINES {
  INPUT = triangle_side_angle_data
  IDENTIFY = valid_opposite_pairs
  SOLVE = unknown_side_or_angle
  IF SSA_angle_recovery
   CHECK = ambiguous_case
  RETURN = triangle_result
 }

 R7_AREA {
  INPUT = [a,b,C]
  REQUIRE = included_angle_C
  COMPUTE = (1/2)*a*b*sin(C)
  RETURN = area
 }

 R8_GRAPH_REASONING {
  INPUT = y=A*sin(Bx+C)+D
  IDENTIFY = amplitude=abs(A)
  IDENTIFY = period=2*pi/abs(B) WHEN B!=0
  IDENTIFY = phase_shift=-C/B WHEN B!=0
  IDENTIFY = vertical_shift=D
  RETURN = graph_parameters
 }

 R9_VECTOR_RELATION {
  INPUT = [u,v,theta]
  RELATE = |u×v|=|u||v|sin(theta) WHEN 3D_or_area_context
  RETURN = sine_based_vector_measure
 }
}

CONSTRAINTS {
 C1 = "Right-triangle sine definition uses opposite side relative to the chosen angle."
 C2 = "Hypotenuse must be the side opposite the right angle."
 C3 = "Real sine values lie in [-1,1]."
 C4 = "Square-root recovery from sin^2 requires sign information."
 C5 = "arcsin gives a principal value, not all solutions."
 C6 = "Law of Sines angle recovery may have an ambiguous second solution."
 C7 = "Angle units must be consistent."
 C8 = "Hyperbolic sine sinh must not be conflated with sin."
 C9 = "The notation sin^-1 means inverse sine in this ontology, not reciprocal cosecant."
}

SEMANTIC_EQUIVALENCE {
 "Sine" <=> "UnitCircleYCoordinate"
 "Sine" <=> "OppositeOverHypotenuse" WHEN right_triangle_context
 "InverseSine" <=> "Arcsine"
 "SineGraph" <=> "PeriodicWaveYEqualsSinX"
 "OddFunctionSine" <=> "sin(-x)=-sin(x)"
}

VALIDATION {
 V1 = "Right-triangle ratio uses opposite/hypotenuse."
 V2 = "Unit-circle interpretation uses y-coordinate."
 V3 = "Range is [-1,1]."
 V4 = "Period is 2*pi."
 V5 = "Odd symmetry holds."
 V6 = "Pythagorean identity is consistent."
 V7 = "arcsin input lies in [-1,1]."
 V8 = "Law of Sines preserves opposite side-angle pairing."
 V9 = "SSA ambiguity is checked when relevant."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SINE_DEFINITION = 0.99
 RIGHT_TRIANGLE_REASONING = 0.99
 UNIT_CIRCLE_REASONING = 0.99
 IDENTITY_REASONING = 0.99
 INVERSE_SINE_REASONING = 0.98
 LAW_OF_SINES_REASONING = 0.99
 GRAPH_REASONING = 0.99

 POLICY =
  "Reduce confidence when angle units, quadrant, principal-value convention, side-angle correspondence or SSA ambiguity are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SineReasoning,
  TrigonometryReasoning,
  RightTriangleReasoning,
  UnitCircleReasoning,
  IdentityReasoning,
  InverseTrigReasoning,
  TriangleSolving,
  GraphReasoning,
  VectorReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "sine" -> Sine
  "sin" -> Sine
  "arcsin" -> InverseSine
  "inverse sine" -> InverseSine
  "sin^-1" -> InverseSine WHEN inverse_function_context
  "sine rule" -> LawOfSines
  "law of sines" -> LawOfSines
 }

 INTERPRETATION_ORDER = [
  detect_sine_context,
  identify right_triangle_unit_circle_identity_graph_triangle_or_vector context,
  identify angle_units_and_known_values,
  choose ratio_coordinate_identity_inverse_or_law_of_sines pathway,
  compute sine_value_angle_side_area_or_graph_parameter,
  check range_quadrant_domain_and_ambiguity constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("sine")
  OR query_mentions("sin")
  THEN activate(SineOntology)

 E2:
  IF right_triangle_context
  THEN use opposite/hypotenuse

 E3:
  IF arbitrary_angle_context
  THEN use unit_circle_definition

 E4:
  IF inverse_sine_requested
  THEN require input_in_[-1,1]
  AND return principal_value unless broader solution set is requested

 E5:
  IF Pythagorean_identity_context
  THEN use sin^2(theta)=1-cos^2(theta)
  AND resolve sign from quadrant_or_context

 E6:
  IF Law_of_Sines_context
  THEN preserve opposite_side_angle_pairs

 E7:
  IF SSA_angle_case
  THEN test ambiguous_second_solution

 E8:
  IF sine_graph_context
  THEN derive amplitude_period_phase_and_vertical_shift
}

RECURSION {
 ENTRY = Sine

 EXPAND(Sine) -> [
  RightTriangleSine,
  UnitCircleSine,
  SineGraph,
  InverseSine,
  Periodicity,
  OddFunctionSymmetry,
  PythagoreanIdentity,
  LawOfSines,
  TriangleAreaFormula
 ]

 EXPAND(RightTriangleSine) -> [
  OppositeSide,
  Hypotenuse,
  Angle,
  SimilarTriangles
 ]

 EXPAND(UnitCircleSine) -> [
  YCoordinate,
  Quadrant,
  ReferenceAngle,
  Periodicity,
  Cosine
 ]

 EXPAND(InverseSine) -> [
  PrincipalDomain,
  PrincipalRange,
  GeneralSolutions,
  AmbiguousTriangleCase
 ]

 STOP_CONDITION =
  "Stop when the sine value, angle, triangle side, identity, graph parameter, vector relation or geometric application is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Trigonometric Geometry"

 IMMEDIATE_CONTEXTS = [
  "Trigonometry Ontology",
  "Cosine Ontology",
  "Tangent Ontology",
  "Triangle Ontology",
  "Circle Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Trigonometric Geometry",
  "Trigonometry Ontology",
  "Cosine Ontology",
  "Tangent Ontology",
  "Triangle Ontology",
  "Circle Ontology",
  "Angle Ontology",
  "Coordinate Geometry Ontology",
  "Vector Geometry Ontology",
  "Scientific Measurement Ontologies"
 ]

 ROUTE_RULES = [
  "Sine -> Trigonometry Ontology for parent function framework",
  "Sine -> Triangle for right-triangle and Law-of-Sines reasoning",
  "Sine -> Circle for unit-circle semantics",
  "Sine -> Cosine for Pythagorean identity",
  "Sine -> Tangent for quotient identities",
  "Sine -> Angle for angle measurement",
  "Sine -> Coordinate Geometry for sine graphs",
  "Sine -> Vector Geometry for cross-product magnitude and area relations"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve right-triangle opposite/hypotenuse semantics.",
  "Preserve unit-circle y-coordinate semantics.",
  "Preserve range [-1,1].",
  "Preserve odd symmetry and 2*pi periodicity.",
  "Preserve inverse-sine principal-value semantics.",
  "Preserve Law-of-Sines correspondence.",
  "Preserve machine-readable sine semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Sine is the trigonometric function equal to opposite/hypotenuse in a right triangle and to the y-coordinate on the unit circle."

 RIGHT_TRIANGLE_SUMMARY =
  "sin(theta)=opposite/hypotenuse"

 UNIT_CIRCLE_SUMMARY =
  "P(theta)=(cos(theta),sin(theta)); sine is the y-coordinate."

 RANGE_SUMMARY =
  "-1<=sin(theta)<=1"

 PERIOD_SUMMARY =
  "sin(theta+2*pi)=sin(theta)"

 SYMMETRY_SUMMARY =
  "sin(-theta)=-sin(theta)"

 INVERSE_SUMMARY =
  "arcsin:[-1,1]->[-pi/2,pi/2]"

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

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

FLOW {
 INPUT
 -> detect_sine_context
 -> identify triangle_unit_circle_identity_graph_inverse_law_of_sines_or_vector context
 -> identify angle_units_known_values_and_target
 -> select sine_ratio_coordinate_identity_inverse_or_triangle pathway
 -> perform sine_specific reasoning
 -> verify range_quadrant_side_role_domain_principal_value_and_ambiguity constraints
 -> route to trigonometry triangle circle cosine tangent angle coordinate vector_or_measurement ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Sine Ontology defines sine through right-triangle ratio and unit-circle coordinate interpretations. It supports periodicity, odd symmetry, inverse sine, Pythagorean identities, the Law of Sines, triangle area, coordinate graphs, reference-angle reasoning and vector applications while preserving angle, domain and ambiguity constraints."
}

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

}

END_FLOW