{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.trigonometric-function","code":"thing-q93344","url":"https://ver.cy/models/thing/q93344/","name":"trigonometric function","alternateNames":["cas function","chord function","tangent and cotangent","sine and cosine","secant and cosecant","rarely used trigonometric functions","Jyā, koti-jyā and utkrama-jyā"],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"The functions of an angle that relate the angles of a right triangle to ratios of its sides and extend through the unit circle beyond acute angles, 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; sine and cosine are defined for every real input, tangent and secant are undefined where cosine is zero, and cotangent and cosecant are undefined where sine is zero. Trigonometric functions are periodic, elementary and transcendental, and are fundamental in geometry, physics, signal processing and engineering.","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.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Unit circle definition","Periodicity","Rich identities","Fourier basis"],"structure":{"bundles":[{"id":"define","name":"Define","description":"Definitions.","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.","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.","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.","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"}]}]}]}]},"agentConduct":{"may":["Compute sine, cosine, tangent and related functions and state whether angles are in degrees or radians.","Explain identities and inverse functions with their valid ranges."],"mustNot":["Mix degrees and radians silently in a calculation.","Return an inverse value outside its principal range without saying so.","Present a rounded value as exact in navigation, surveying or engineering results.","Hide a division by zero, such as tangent at a right angle, behind a large number."],"requiresHuman":["Relying on its own trigonometric calculations for structural, navigation or safety-critical work."]},"ethics":{"considerations":["Angle errors in surveying, navigation and construction can cause accidents and boundary disputes.","Students rely on worked answers; wrong ones mislead learning."],"affectedParties":["Engineers and surveyors","Students","People relying on computed positions"]},"owners":{"steward":"The public: held in common as mathematical knowledge.","roles":[],"masterSystems":[]},"relations":[{"target":"Q184743","type":"parent","note":"registry parent class"},{"target":"Q824282","type":"parent","note":"registry parent class"},{"target":"Q864333","type":"parent","note":"registry parent class"},{"target":"periodic function","type":"related","note":"in registry terms"},{"target":"elementary function","type":"related","note":"in registry terms"},{"target":"transcendental function","type":"related","note":"in registry terms"},{"target":"unit circle","type":"related","note":"for all real arguments"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.trigonometric-function","Wikidata: Q93344 (https://www.wikidata.org/wiki/Q93344)"]},"properties":{"applicability":"not-applicable","items":["period of sine: 2 pi radians","principal functions: 6 count"]},"recognition":{"applicability":"optional","items":["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","Wave-shaped graphs of sine and cosine and the unit circle diagram."]},"capabilities":{"applicability":"required","items":["explain definitions and relations","relay identities and values","describe applications","distinguish related functions"]},"hazards":{"applicability":"required","items":["Degree and radian confusion","Notation ambiguities such as sin squared and inverse","Domain errors for tangent and secant"]},"interfaces":{"applicability":"required","items":["ISO 80000-2 notation for trigonometric functions","No regulation"]},"context":{"applicability":"required","items":["Relating angles to lengths and modelling periodic phenomena.","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"]}},"sources":[{"title":"Wikidata item Q93344: trigonometric function","url":"https://www.wikidata.org/wiki/Q93344","note":"identity and sense of the item"},{"title":"Wikipedia: Trigonometric functions","url":"https://en.wikipedia.org/wiki/Trigonometric_functions","note":"general description of the item"}],"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?"],"resources":{"spec":"/models/things/publications/thing-q93344/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q93344/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T01:51:10Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:33+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}