Inverse Trigonometric Function Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of inverse trigonometric functions within the Kireth Research Semantic Library. Inverse trigonometric functions recover principal angle values from trigonometric ratios through restricted-domain inverses of sine, cosine, tangent and related functions. This ontology supports arcsine, arccosine, arctangent, principal domains and ranges, branch restrictions, triangle-angle recovery, unit-circle interpretation, exact values, equation solving, derivative and analytic links where relevant, ambiguity handling and relationships with sine, cosine and tangent ontologies.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = InverseTrigonometricFunction
 SYMBOLIC_FORMS = [
  "arcsin(x)",
  "arccos(x)",
  "arctan(x)",
  "sin^-1(x)",
  "cos^-1(x)",
  "tan^-1(x)"
 ]
 ENTITY_CLASS = InverseFunctionFamily
 PRIMARY_CONTEXT = Trigonometry
 DEFINITION = "An inverse trigonometric function returns a principal angle whose trigonometric function equals a specified value, using restricted domains to make the original trigonometric function one-to-one."
 SEMANTIC_ROLE = "Provides the principal-angle recovery framework for trigonometric ratios, triangle solving, equations, unit-circle reasoning, branch control and exact inverse-function semantics."
}

SCOPE {
 INCLUDE = [
  InverseTrigonometricFunction,
  Arcsine,
  InverseSine,
  Arccosine,
  InverseCosine,
  Arctangent,
  InverseTangent,
  Arccotangent,
  Arcsecant,
  Arccosecant,
  PrincipalValue,
  PrincipalDomain,
  PrincipalRange,
  RestrictedDomain,
  Branch,
  AngleRecovery,
  TrigonometricEquation,
  TriangleAngle,
  UnitCircle,
  ExactValue
 ]

 EXCLUDE_AS_PRIMARY = [
  ReciprocalTrigonometricFunctionWhenInverseMeaningIsNotIntended,
  HyperbolicInverseFunctionsUnlessExplicit,
  ComplexBranchTheoryUnlessExplicit,
  GeneralFunctionalInverseWithoutTrigonometricContext
 ]

 DISAMBIGUATION_RULE =
  "When arcsin, arccos, arctan, sin^-1, cos^-1 or tan^-1 is used to recover an angle from a trigonometric value, interpret it as an inverse trigonometric function rather than a reciprocal function."
}

CORE_CONCEPTS {

 InverseTrigonometricFunction {
  definition = "A principal inverse obtained by restricting a trigonometric function to a one-to-one interval."
 }

 Arcsine {
  symbol = "arcsin"
  alternative = "sin^-1"
  domain = "[-1,1]"
  principal_range = "[-pi/2,pi/2]"
  inverse_relation = "sin(arcsin(x))=x for x in [-1,1]"
 }

 Arccosine {
  symbol = "arccos"
  alternative = "cos^-1"
  domain = "[-1,1]"
  principal_range = "[0,pi]"
  inverse_relation = "cos(arccos(x))=x for x in [-1,1]"
 }

 Arctangent {
  symbol = "arctan"
  alternative = "tan^-1"
  domain = "R"
  principal_range = "(-pi/2,pi/2)"
  inverse_relation = "tan(arctan(x))=x for x in R"
 }

 Arccotangent {
  definition = "An inverse cotangent function whose principal range depends on convention."
 }

 Arcsecant {
  definition = "An inverse secant function with domain |x|>=1 and convention-dependent principal range."
 }

 Arccosecant {
  definition = "An inverse cosecant function with domain |x|>=1 and convention-dependent principal range."
 }

 PrincipalValue {
  definition = "The unique output selected from the principal range of an inverse trigonometric function."
 }

 RestrictedDomain {
  definition = "A chosen interval on which a periodic trigonometric function is one-to-one."
 }

 Branch {
  definition = "A single-valued inverse selection from a multivalued inverse relation."
 }
}

RELATIONS {
 InverseTrigonometricFunction --is_inverse_of_restricted--> TrigonometricFunction
 Arcsine --is_inverse_of_restricted--> Sine
 Arccosine --is_inverse_of_restricted--> Cosine
 Arctangent --is_inverse_of_restricted--> Tangent
 Arcsine --returns--> PrincipalValue
 Arccosine --returns--> PrincipalValue
 Arctangent --returns--> PrincipalValue
 InverseTrigonometricFunction --uses--> RestrictedDomain
 RestrictedDomain --determines--> PrincipalRange
 InverseTrigonometricFunction --supports--> AngleRecovery
 InverseTrigonometricFunction --supports--> TriangleSolving
 InverseTrigonometricFunction --supports--> TrigonometricEquation
 UnitCircle --supports--> PrincipalAngleInterpretation
}

OPERATORS {
 ARCSIN(x) :=
  unique theta in [-pi/2,pi/2]
  such_that sin(theta)=x
  WHEN -1<=x<=1

 ARCCOS(x) :=
  unique theta in [0,pi]
  such_that cos(theta)=x
  WHEN -1<=x<=1

 ARCTAN(x) :=
  unique theta in (-pi/2,pi/2)
  such_that tan(theta)=x

 SIN_OF_ARCSIN(x) :=
  x
  WHEN x in [-1,1]

 COS_OF_ARCCOS(x) :=
  x
  WHEN x in [-1,1]

 TAN_OF_ARCTAN(x) :=
  x
  WHEN x in R

 ARCSIN_OF_SIN(theta) :=
  principal_representative_of_theta_in_sine_branch

 ARCCOS_OF_COS(theta) :=
  principal_representative_of_theta_in_cosine_branch

 ARCTAN_OF_TAN(theta) :=
  principal_representative_of_theta_mod_pi_in_tangent_branch

 ANGLE_FROM_RIGHT_TRIANGLE_SINE(opposite,hypotenuse) :=
  arcsin(opposite/hypotenuse)

 ANGLE_FROM_RIGHT_TRIANGLE_COSINE(adjacent,hypotenuse) :=
  arccos(adjacent/hypotenuse)

 ANGLE_FROM_RIGHT_TRIANGLE_TANGENT(opposite,adjacent) :=
  arctan(opposite/adjacent)
}

DEFINITIONS {
 D1 = "Inverse trigonometric functions require restricted domains because sine, cosine and tangent are periodic."
 D2 = "arcsin has real domain [-1,1] and principal range [-pi/2,pi/2]."
 D3 = "arccos has real domain [-1,1] and principal range [0,pi]."
 D4 = "arctan has real domain R and principal range (-pi/2,pi/2)."
 D5 = "sin(arcsin(x))=x for x in [-1,1]."
 D6 = "cos(arccos(x))=x for x in [-1,1]."
 D7 = "tan(arctan(x))=x for all real x."
 D8 = "arcsin(sin(theta)) is not always equal to theta."
 D9 = "arccos(cos(theta)) is not always equal to theta."
 D10 = "arctan(tan(theta)) returns the principal representative modulo pi."
}

AXIOMS {
 A1_RESTRICTED_INVERTIBILITY:
  A trigonometric function becomes invertible only after restriction to a one-to-one interval

 A2_ARCSIN_BRANCH:
  Sine restricted to [-pi/2,pi/2] is one-to-one and onto [-1,1]

 A3_ARCCOS_BRANCH:
  Cosine restricted to [0,pi] is one-to-one and onto [-1,1]

 A4_ARCTAN_BRANCH:
  Tangent restricted to (-pi/2,pi/2) is one-to-one and onto R

 A5_RIGHT_INVERSE:
  f(f^-1(x))=x on the inverse function domain

 A6_LEFT_INVERSE_ON_BRANCH:
  f^-1(f(theta))=theta when theta lies inside the chosen principal branch

 A7_PRINCIPAL_VALUE:
  Inverse trigonometric notation returns a principal value unless a broader solution set is explicitly requested

 A8_PERIODIC_PREIMAGE:
  General trigonometric equations require reconstruction of all periodic preimages from the principal solution
}

REASONING {

 R1_ARCSIN {
  INPUT = x
  REQUIRE = -1<=x<=1
  COMPUTE = principal_theta=arcsin(x)
  RETURN = theta in [-pi/2,pi/2]
 }

 R2_ARCCOS {
  INPUT = x
  REQUIRE = -1<=x<=1
  COMPUTE = principal_theta=arccos(x)
  RETURN = theta in [0,pi]
 }

 R3_ARCTAN {
  INPUT = x
  COMPUTE = principal_theta=arctan(x)
  RETURN = theta in (-pi/2,pi/2)
 }

 R4_TRIANGLE_ANGLE_RECOVERY {
  INPUT = right_triangle_side_ratio
  IDENTIFY = sine_cosine_or_tangent_relation
  APPLY = matching_inverse_function
  VERIFY = triangle_angle_range
  RETURN = angle
 }

 R5_GENERAL_SINE_EQUATION {
  INPUT = sin(theta)=a
  REQUIRE = -1<=a<=1
  COMPUTE = alpha=arcsin(a)
  RETURN = theta=alpha+2k*pi OR theta=pi-alpha+2k*pi
  FOR integer k
 }

 R6_GENERAL_COSINE_EQUATION {
  INPUT = cos(theta)=a
  REQUIRE = -1<=a<=1
  COMPUTE = alpha=arccos(a)
  RETURN = theta=±alpha+2k*pi
  FOR integer k
 }

 R7_GENERAL_TANGENT_EQUATION {
  INPUT = tan(theta)=a
  COMPUTE = alpha=arctan(a)
  RETURN = theta=alpha+k*pi
  FOR integer k
 }

 R8_COMPOSITION_SIMPLIFICATION {
  INPUT = inverse_trig_composed_with_trig_or_trig_with_inverse
  IDENTIFY = composition_order
  APPLY = exact_inverse_only_when_domain_branch_conditions_are_met
  OTHERWISE = principalise_angle
  RETURN = simplified_result
 }

 R9_EXACT_VALUE {
  INPUT = special_ratio
  MATCH = known_unit_circle_or_special_triangle_value
  RETURN = exact_principal_angle
 }
}

CONSTRAINTS {
 C1 = "sin^-1, cos^-1 and tan^-1 denote inverse functions here, not reciprocals."
 C2 = "arcsin and arccos require real input in [-1,1]."
 C3 = "Principal values must stay within the specified principal ranges."
 C4 = "Inverse trigonometric composition is branch-sensitive."
 C5 = "General equation solving requires periodic solution families beyond the principal value."
 C6 = "Angle units must be explicit or consistently inferred."
 C7 = "Right-triangle angle recovery must respect the geometric angle range."
 C8 = "Arcsecant, arccosecant and arccotangent conventions may differ between sources."
 C9 = "Complex inverse-trigonometric branches require separate handling."
}

SEMANTIC_EQUIVALENCE {
 "InverseSine" <=> "Arcsine"
 "InverseCosine" <=> "Arccosine"
 "InverseTangent" <=> "Arctangent"
 "PrincipalValue" <=> "BranchSelectedInverseAngle"
 "RestrictedDomain" <=> "OneToOneTrigBranch"
 "sin^-1(x)" <=> "arcsin(x)" WHEN inverse_function_context
 "cos^-1(x)" <=> "arccos(x)" WHEN inverse_function_context
 "tan^-1(x)" <=> "arctan(x)" WHEN inverse_function_context
}

VALIDATION {
 V1 = "arcsin input lies in [-1,1]."
 V2 = "arccos input lies in [-1,1]."
 V3 = "arctan accepts all real inputs."
 V4 = "arcsin output lies in [-pi/2,pi/2]."
 V5 = "arccos output lies in [0,pi]."
 V6 = "arctan output lies in (-pi/2,pi/2)."
 V7 = "Composition simplification respects branch restrictions."
 V8 = "General equation solutions include periodic families."
 V9 = "Inverse notation is distinguished from reciprocal notation."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 INVERSE_TRIG_DEFINITION = 0.99
 ARCSIN_REASONING = 0.99
 ARCCOS_REASONING = 0.99
 ARCTAN_REASONING = 0.99
 BRANCH_REASONING = 0.99
 TRIANGLE_ANGLE_REASONING = 0.99
 GENERAL_EQUATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when angle units, branch convention, inverse-versus-reciprocal notation, complex domain or triangle-angle context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  InverseTrigonometricReasoning,
  PrincipalValueReasoning,
  BranchReasoning,
  TriangleAngleRecovery,
  TrigonometricEquationSolving,
  UnitCircleReasoning,
  ExactValueReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "inverse trigonometric function" -> InverseTrigonometricFunction
  "inverse trig" -> InverseTrigonometricFunction
  "arcsin" -> Arcsine
  "inverse sine" -> Arcsine
  "sin^-1" -> Arcsine WHEN inverse_function_context
  "arccos" -> Arccosine
  "inverse cosine" -> Arccosine
  "cos^-1" -> Arccosine WHEN inverse_function_context
  "arctan" -> Arctangent
  "inverse tangent" -> Arctangent
  "tan^-1" -> Arctangent WHEN inverse_function_context
 }

 INTERPRETATION_ORDER = [
  detect_inverse_trigonometric_context,
  identify inverse_function_type,
  identify input_domain_and_angle_units,
  determine whether principal_value_or_general_solution_is_requested,
  apply correct inverse_branch,
  reconstruct periodic_solution_family_when_needed,
  verify domain_range_and_branch constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("inverse trigonometric")
  OR query_mentions("arcsin")
  OR query_mentions("arccos")
  OR query_mentions("arctan")
  THEN activate(InverseTrigonometricFunctionOntology)

 E2:
  IF arcsin_requested
  THEN require input_in_[-1,1]
  AND return principal_value_in_[-pi/2,pi/2]

 E3:
  IF arccos_requested
  THEN require input_in_[-1,1]
  AND return principal_value_in_[0,pi]

 E4:
  IF arctan_requested
  THEN return principal_value_in_(-pi/2,pi/2)

 E5:
  IF inverse_function_composed_after_matching_trig_function
  THEN simplify_to_original_angle_only_if_angle_lies_in_principal_branch

 E6:
  IF matching_trig_function_composed_after_inverse
  THEN simplify_to_input_on_inverse_domain

 E7:
  IF general_trigonometric_equation_requested
  THEN expand principal_solution_into_periodic_family

 E8:
  IF right_triangle_angle_recovery
  THEN choose arcsin_arccos_or_arctan from available_side_ratio
}

RECURSION {
 ENTRY = InverseTrigonometricFunction

 EXPAND(InverseTrigonometricFunction) -> [
  Arcsine,
  Arccosine,
  Arctangent,
  Arccotangent,
  Arcsecant,
  Arccosecant,
  PrincipalValue,
  RestrictedDomain,
  PrincipalRange,
  Branch,
  AngleRecovery,
  TrigonometricEquation
 ]

 EXPAND(Arcsine) -> [
  DomainMinusOneToOne,
  PrincipalRangeMinusPiOverTwoToPiOverTwo,
  SineOntology,
  GeneralSineEquation
 ]

 EXPAND(Arccosine) -> [
  DomainMinusOneToOne,
  PrincipalRangeZeroToPi,
  CosineOntology,
  GeneralCosineEquation
 ]

 EXPAND(Arctangent) -> [
  RealDomain,
  PrincipalRangeMinusPiOverTwoToPiOverTwoOpen,
  TangentOntology,
  GeneralTangentEquation
 ]

 EXPAND(BranchReasoning) -> [
  PrincipalValue,
  Periodicity,
  ReferenceAngle,
  GeneralSolutions
 ]

 STOP_CONDITION =
  "Stop when the principal angle, branch, triangle angle, inverse composition or full periodic solution set is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Trigonometric Geometry"

 IMMEDIATE_CONTEXTS = [
  "Trigonometry Ontology",
  "Sine Ontology",
  "Cosine Ontology",
  "Tangent Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Trigonometric Geometry",
  "Trigonometry Ontology",
  "Sine Ontology",
  "Cosine Ontology",
  "Tangent Ontology",
  "Triangle Ontology",
  "Circle Ontology",
  "Angle Ontology",
  "Coordinate Geometry Ontology",
  "Analytic Geometry Ontology"
 ]

 ROUTE_RULES = [
  "Inverse Trigonometric Function -> Trigonometry Ontology for parent inverse-function framework",
  "Inverse Trigonometric Function -> Sine for arcsine",
  "Inverse Trigonometric Function -> Cosine for arccosine",
  "Inverse Trigonometric Function -> Tangent for arctangent",
  "Inverse Trigonometric Function -> Triangle for angle recovery",
  "Inverse Trigonometric Function -> Circle for unit-circle branch interpretation",
  "Inverse Trigonometric Function -> Angle for angular measure and principal ranges",
  "Inverse Trigonometric Function -> Analytic Geometry for inverse-function graphs and equations"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible inverse-function branches, conventions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve restricted-domain inverse semantics.",
  "Preserve arcsin, arccos and arctan principal ranges.",
  "Preserve inverse-versus-reciprocal distinction.",
  "Preserve branch-sensitive composition.",
  "Preserve periodic reconstruction of general solutions.",
  "Preserve domain restrictions.",
  "Preserve machine-readable inverse-trigonometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An inverse trigonometric function returns a principal angle corresponding to a specified trigonometric value after restricting the original function to a one-to-one domain."

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

 ARCCOS_SUMMARY =
  "arccos:[-1,1]->[0,pi]"

 ARCTAN_SUMMARY =
  "arctan:R->(-pi/2,pi/2)"

 COMPOSITION_SUMMARY =
  "f(f^-1(x))=x on the inverse domain; f^-1(f(theta))=theta only on the chosen principal branch."

 SINE_EQUATION_SUMMARY =
  "sin(theta)=a gives theta=arcsin(a)+2k*pi or theta=pi-arcsin(a)+2k*pi."

 COSINE_EQUATION_SUMMARY =
  "cos(theta)=a gives theta=±arccos(a)+2k*pi."

 TANGENT_EQUATION_SUMMARY =
  "tan(theta)=a gives theta=arctan(a)+k*pi."
}

FLOW {
 INPUT
 -> detect_inverse_trigonometric_context
 -> identify arcsin_arccos_arctan_or_other_inverse
 -> identify input_domain_angle_units_and_geometric_context
 -> decide principal_value_or_general_solution goal
 -> apply correct restricted_inverse_branch
 -> reconstruct periodic_preimages_when_required
 -> verify domain_range_branch_notation_and_triangle constraints
 -> route to trigonometry sine cosine tangent triangle circle angle_or_analytic_geometry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Inverse Trigonometric Function Ontology defines arcsine, arccosine, arctangent and related inverse functions as branch-restricted inverses of periodic trigonometric functions. It supports principal-value angle recovery, triangle solving, exact values, branch-sensitive composition and reconstruction of full periodic solution sets while explicitly distinguishing inverse notation from reciprocal trigonometric functions."
}

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

}

END_FLOW