trigonometric function
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.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
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.
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.
It can be explain definitions and relations; relay identities and values; describe applications; distinguish related functions.
Distinguishing features
Unit circle definition
Periodicity
Rich identities
Fourier basis
What it looks like
Wave-shaped graphs of sine and cosine and the unit circle diagram.
Physical character
period of sine: 2 pi radians
principal functions: 6 count
How it is recognised
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
Related models
is a kind of - in registry terms
is a kind of - in registry terms
is a kind of - in registry terms
is defined on - for all real arguments
In practice
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
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
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Define Definitions.
Mathematics.
Definition
Definitions.
Definition
Definition.
- How are the trigonometric functions defined by triangles, the unit circle and series? definition
- Is the question about trigonometric, inverse or hyperbolic functions? boundary
Relations
Identities.
Relations
Relations.
- What are the key identities, including Pythagorean, angle sum and double angle formulas? definition
- Which entry fits trigonometric identity? action
Compute Values and computation.
Practice.
Values
Special values and graphs.
Values
Values.
- What are the special values and graph features of each function? measurement
- Which references are standard? provenance
Algorithms
Computation.
Algorithms
Algorithms.
- How are trigonometric functions computed by series, CORDIC and tables? provenance
- Which entry fits CORDIC? action
Apply Applications.
Application.
Physics
Physics and engineering.
Physics
Physics.
- How are trigonometric functions used in waves, rotations, navigation and signal processing? provenance
- Which entry fits Fourier analysis? action
Historical
Historical and rare functions.
Historical
Historical.
- What were the chord, versine and haversine used for, and what is the cas function? provenance
- Which sources are cited? provenance
Context History and teaching.
Context.
History
History.
History
History.
- How did trigonometry develop from Hipparchus and Indian sine tables to Euler? provenance
- Which entry fits the history of trigonometry? action
Teaching
Teaching.
Teaching
Teaching.
- How are trigonometric functions taught, and what misconceptions arise? provenance
- Which entry fits mathematics education? action
What the second pass must settle
- 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?