{ "model": { "rank": 3245, "code": "thing-q93344", "model_id": "vr.tr.trigonometric-function", "name": "trigonometric function", "purpose": "Let an agent explain the trigonometric functions and their relations, relay identities, values and applications from mathematical references, describe historical and rarely used functions, and distinguish trigonometric from hyperbolic and inverse functions.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q93344/", "tier": 2, "score": 89, "payload": { "layer": "wikidata", "aliases": [ "cas function", "chord function", "tangent and cotangent", "sine and cosine", "secant and cosecant", "rarely used trigonometric functions", "Jyā, koti-jyā and utkrama-jyā" ], "aliasCount": 7, "merged": 7, "knownIn": 89, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q93344", "url": "https://www.wikidata.org/wiki/Q93344", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T01:51:10Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q93344", "registryId": "vr.tr.trigonometric-function", "name": "trigonometric function", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q93344/", "model": { "registry_id": "vr.tr.trigonometric-function", "name": "trigonometric function", "purpose": "Let an agent explain the trigonometric functions and their relations, relay identities, values and applications from mathematical references, describe historical and rarely used functions, and distinguish trigonometric from hyperbolic and inverse functions.", "definition": "The functions of an angle that relate the angles of a right triangle to ratios of its sides and extend to all real numbers through the unit circle, principally sine and cosine, tangent and cotangent, secant and cosecant, with historically used functions such as the chord, versine and haversine, and the cas function of the Hartley transform; trigonometric functions are periodic, elementary and transcendental, and are fundamental in geometry, physics, signal processing and engineering.", "what_it_is_for": "Relating angles to lengths and modelling periodic phenomena.", "affordances": [ "explain definitions and relations", "relay identities and values", "describe applications", "distinguish related functions" ], "distinguishing_features": [ "Unit circle definition", "Periodicity", "Rich identities", "Fourier basis" ], "appearance": "Wave-shaped graphs of sine and cosine and the unit circle diagram.", "visual_identification": [ "Functions of angles based on the unit circle", "Sine, cosine, tangent, cotangent, secant, cosecant and historical functions", "Hyperbolic functions use the hyperbola; inverse trigonometric functions return angles" ], "physical_properties": [ { "quantity": "period of sine", "typical_range": "2 pi", "unit": "radians", "note": "" }, { "quantity": "principal functions", "typical_range": "6", "unit": "count", "note": "" } ], "families_and_kinds": [ "sine and cosine", "tangent and cotangent", "secant and cosecant", "historical functions such as chord, versine, haversine and exsecant", "the cas function", "inverse trigonometric functions as related", "hyperbolic analogues as related" ], "related_models": [ { "relation": "is a kind of", "target": "periodic function", "why": "in registry terms" }, { "relation": "is a kind of", "target": "elementary function", "why": "in registry terms" }, { "relation": "is a kind of", "target": "transcendental function", "why": "in registry terms" }, { "relation": "is defined on", "target": "unit circle", "why": "for all real arguments" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation for trigonometric functions", "No regulation" ], "failure_modes_and_hazards": [ "Degree and radian confusion", "Notation ambiguities such as sin squared and inverse", "Domain errors for tangent and secant" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "Definitions.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definitions.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "How are the trigonometric functions defined by triangles, the unit circle and series?", "kind": "definition" }, { "text": "Is the question about trigonometric, inverse or hyperbolic functions?", "kind": "boundary" } ] } ] }, { "id": "relations", "name": "Relations", "description": "Identities.", "findings": [ { "id": "relations-finding", "name": "Relations", "description": "Relations.", "questions": [ { "text": "What are the key identities, including Pythagorean, angle sum and double angle formulas?", "kind": "definition" }, { "text": "Which entry fits trigonometric identity?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Values and computation.", "rationale": "Practice.", "layers": [ { "id": "values", "name": "Values", "description": "Special values and graphs.", "findings": [ { "id": "values-finding", "name": "Values", "description": "Values.", "questions": [ { "text": "What are the special values and graph features of each function?", "kind": "measurement" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "algorithms", "name": "Algorithms", "description": "Computation.", "findings": [ { "id": "algorithms-finding", "name": "Algorithms", "description": "Algorithms.", "questions": [ { "text": "How are trigonometric functions computed by series, CORDIC and tables?", "kind": "provenance" }, { "text": "Which entry fits CORDIC?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Application.", "layers": [ { "id": "physics", "name": "Physics", "description": "Physics and engineering.", "findings": [ { "id": "physics-finding", "name": "Physics", "description": "Physics.", "questions": [ { "text": "How are trigonometric functions used in waves, rotations, navigation and signal processing?", "kind": "provenance" }, { "text": "Which entry fits Fourier analysis?", "kind": "action" } ] } ] }, { "id": "historical", "name": "Historical", "description": "Historical and rare functions.", "findings": [ { "id": "historical-finding", "name": "Historical", "description": "Historical.", "questions": [ { "text": "What were the chord, versine and haversine used for, and what is the cas function?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and teaching.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did trigonometry develop from Hipparchus and Indian sine tables to Euler?", "kind": "provenance" }, { "text": "Which entry fits the history of trigonometry?", "kind": "action" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Teaching.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How are trigonometric functions taught, and what misconceptions arise?", "kind": "provenance" }, { "text": "Which entry fits mathematics education?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should sine and cosine be the primary linked entries?", "How should reference tables be linked?", "The registry entry has merged aliases naming pairs and rare functions; should they be split off?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a trigonometric function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain the trigonometric functions and their relations, relay identities, values and applications from mathematical references, describe historical and rarely used functions, and distinguish trigonometric from hyperbolic and inverse functions.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: cas function, chord function, tangent and cotangent, sine and cosine, secant and cosecant, rarely used trigonometric functions, Jyā, koti-jyā and utkrama-jyā.", "7 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 89 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a trigonometric function, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "cas function", "chord function", "tangent and cotangent", "sine and cosine", "secant and cosecant", "rarely used trigonometric functions", "Jyā, koti-jyā and utkrama-jyā" ] } ], "standing": "Described in 89 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 7 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one trigonometric function from another, and a trigonometric function from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a trigonometric function sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a trigonometric function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a trigonometric function passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a trigonometric function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a trigonometric function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a trigonometric function is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What trigonometric function is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of trigonometric function are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is trigonometric function operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of trigonometric function and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of trigonometric function?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as trigonometric function?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a trigonometric function came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a trigonometric function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a trigonometric function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a trigonometric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a trigonometric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a trigonometric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }