Trigonometric Expression Ontology

FLOW ALGEBRA
TRIGONOMETRIC EXPRESSION ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Expression Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/trigonometric-expression-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Status: mathematical reference model; source page not independently verified

01. PURPOSE
flow "TrigonometricExpressionOntology" v1.0
  input: symbolic expression, assumptions, domain, angle unit
  output: typed expression graph, validated transformations, provenance
  invariant: transformations preserve meaning on explicitly stated domains

02. ENTITY TYPES
  TrigonometricExpression <: MathematicalExpression
  TrigonometricFunction <: Function
  Argument <: MathematicalExpression
  AngleUnit := Radian | Degree
  TrigFunctionKind := sin | cos | tan | cot | sec | csc
  InverseTrigFunctionKind := asin | acos | atan | acot | asec | acsc
  ExpressionNode := FunctionApplication | Sum | Product | Quotient | Power | Constant | Variable

03. CORE SCHEMA
  TrigonometricExpression {
    id: URI
    syntax: string
    ast: -ExpressionNode
    function_kind: TrigFunctionKind?
    argument: MathematicalExpression?
    domain: Set
    codomain: Set
    angle_unit: AngleUnit
    assumptions: Set[Predicate]
    exactness: exact | approximate | symbolic
    provenance: SourceReference?
  }

04. ANGLE CONVENTION
  default_angle_unit = Radian
  radians(degrees) = degrees * pi / 180
  degrees(radians) = radians * 180 / pi
  rule: derivative and limit identities assume radians unless explicitly converted
  rule: never silently mix degree and radian arguments

05. FUNCTION DEFINITIONS (REAL DOMAIN)
  sin(x), cos(x): R -> [-1,1]
  tan(x) = sin(x)/cos(x), where cos(x) != 0
  cot(x) = cos(x)/sin(x), where sin(x) != 0
  sec(x) = 1/cos(x), where cos(x) != 0
  csc(x) = 1/sin(x), where sin(x) != 0
  note: ranges of tan and cot are R on their respective domains
  note: sec and csc have ranges (-infinity,-1] union [1,infinity)

06. FUNDAMENTAL IDENTITIES
  sin(x)^2 + cos(x)^2 = 1
  1 + tan(x)^2 = sec(x)^2, where cos(x) != 0
  1 + cot(x)^2 = csc(x)^2, where sin(x) != 0
  sin(-x) = -sin(x)
  cos(-x) = cos(x)
  tan(-x) = -tan(x), where defined

07. PERIODICITY
  sin(x + 2*pi*k) = sin(x), k in Z
  cos(x + 2*pi*k) = cos(x), k in Z
  tan(x + pi*k) = tan(x), k in Z and defined
  cot(x + pi*k) = cot(x), k in Z and defined

08. ADDITION AND DOUBLE-ANGLE
  sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
  cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
  sin(2*x) = 2*sin(x)*cos(x)
  cos(2*x) = cos(x)^2 - sin(x)^2
  tan(a+b) = (tan(a)+tan(b))/(1-tan(a)*tan(b))
    precondition: tan(a), tan(b), tan(a+b) defined; denominator != 0

09. INVERSE FUNCTIONS
  asin: [-1,1] -> [-pi/2,pi/2]
  acos: [-1,1] -> [0,pi]
  atan: R -> (-pi/2,pi/2)
  sin(asin(y)) = y for y in [-1,1]
  asin(sin(x)) = x only for x in [-pi/2,pi/2]
  note: inverse functions use principal branches; not reciprocals

10. DIFFERENTIATION
  d/dx sin(x) = cos(x)
  d/dx cos(x) = -sin(x)
  d/dx tan(x) = sec(x)^2 where defined
  d/dx sin(g(x)) = cos(g(x))*g'(x)
  d/dx cos(g(x)) = -sin(g(x))*g'(x)
  precondition: g differentiable at evaluation point

11. INTEGRATION
  integral sin(x) dx = -cos(x) + C
  integral cos(x) dx = sin(x) + C
  integral sec(x)^2 dx = tan(x) + C on connected domain components
  note: antiderivatives apply on intervals where integrands are defined

12. EQUATION SOLVING
  sin(x) = 0 => x = k*pi, k in Z
  cos(x) = 0 => x = pi/2 + k*pi, k in Z
  tan(x) = 0 => x = k*pi, k in Z
  sin(x) = sin(a) => x = a + 2*pi*k OR x = pi-a + 2*pi*k
  cos(x) = cos(a) => x = 2*pi*k +/- a
  rule: intersect solutions with the declared domain

13. SEMANTIC RELATIONSHIPS
  TrigonometricExpression --is_a--> MathematicalExpression
  TrigonometricExpression --applies--> TrigonometricFunction
  TrigonometricExpression --has_argument--> Argument
  TrigonometricExpression --has_domain--> Set
  TrigonometricExpression --has_angle_unit--> AngleUnit
  TrigonometricExpression --equivalent_under--> Assumptions
  TrigonometricExpression --related_to--> ExponentialExpression
  TrigonometricExpression --related_to--> ComplexNumber

14. SYMBOLIC REWRITE RULES
  rewrite sin(x)^2 + cos(x)^2 -> 1
  rewrite sin(-x) -> -sin(x)
  rewrite cos(-x) -> cos(x)
  rewrite tan(x) -> sin(x)/cos(x) IF cos(x) != 0
  rewrite sec(x) -> 1/cos(x) IF cos(x) != 0
  rule: track excluded points even after cancellation
  rule: store both original and rewritten domains

15. COMPLEX EXTENSION
  sin(z) = (exp(i*z)-exp(-i*z))/(2*i)
  cos(z) = (exp(i*z)+exp(-i*z))/2
  z in C
  note: complex inverse trigonometric functions require branch conventions

16. NORMALISATION
  parse -> typecheck -> resolve angle units -> infer domain
  -> canonicalise AST -> apply guarded rewrites -> verify equivalence
  canonicalisation: deterministic operand ordering only for commutative operators
  no simplification may silently enlarge the expression domain

17. KNOWLEDGE GRAPH EXAMPLE
  node: expr:sin_squared_plus_cos_squared
    type: TrigonometricExpression
    syntax: "sin(x)^2 + cos(x)^2"
    domain: R
    angle_unit: Radian
  edge: expr:sin_squared_plus_cos_squared --equivalent_to--> constant:one
  evidence: PythagoreanTrigonometricIdentity

18. INFERENCE
  IF expression matches sin(t)^2+cos(t)^2 AND t real
    THEN equivalent_value = 1
  IF denominator includes cos(t)
    THEN require cos(t) != 0
  IF expression has degree argument
    THEN convert to radians before calculus inference
  IF identity requires branch restriction
    THEN attach branch predicate to inference result

19. VALIDATION
  validate syntax and function arity
  validate argument types and angle units
  validate denominator non-zero constraints
  validate declared domain and inverse branch restrictions
  validate transformation by symbolic proof or domain-aware counterexample testing
  reject unsupported global identities

20. CONFIDENCE
  mathematical_identity_with_proof: 1.00
  verified_symbolic_derivation: 0.99
  numerical_check_only: heuristic; not proof
  confidence applies to evidence quality, not mathematical truth value

21. SEMANTIC_EQUIVALENCE
  equivalent(A,B,D) := for_all x in D: defined(A,x) AND defined(B,x)
    AND value(A,x) = value(B,x)
  domain-sensitive: sin(x)/cos(x) and tan(x) equivalent where cos(x) != 0
  alpha-renaming allowed for bound dummy symbols when capture-free

22. FLOW_HASH
  canonical_payload := canonical_AST + domain + assumptions + angle_unit
    + branch_convention + ontology_version
  FLOW_HASH := SHA256(UTF8(canonical_payload))
  rule: hash is identity/checksum metadata, not a proof of equivalence

23. UPDATE_MODEL
  version: semantic_version
  update triggers: definition change | identity correction | schema revision
  preserve: previous_version, change_log, source_provenance
  breaking change: domain, branch or unit interpretation changed

24. AI_CONSUMPTION
  machine_readable: true
  preferred_output: typed graph + rules + validation trace
  reasoning: preserve units, domains, assumptions and branch constraints
  retrieval_keys: trigonometric expression, sine, cosine, tangent, identities
  prohibited: asserting equality beyond a valid domain
  provenance: Kireth Semantic Library source URL above

25. TEST CASES
  PASS: sin(x)^2+cos(x)^2 -> 1 for real x
  PASS: tan(pi/4) -> 1
  PASS: sin(30 degrees) -> 1/2
  FAIL: tan(pi/2) is defined
  FAIL: asin(sin(pi)) = pi
  FAIL: sin(x)/sin(x) = 1 for all real x
  PASS: sin(x)/sin(x) = 1 where sin(x) != 0

END FLOW ALGEBRA