Pythagorean identity in trigonometry
E518471
The Pythagorean identity in trigonometry is a fundamental relation among sine and cosine functions, stating that for any angle θ, sin²θ + cos²θ = 1.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Pythagorean identity | 1 |
| Pythagorean identity in trigonometry canonical | 1 |
| Pythagorean trigonometric identity | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425553 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pythagorean identity in trigonometry Context triple: [Pythagorean theorem, generalizedBy, Pythagorean identity in trigonometry]
-
A.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
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B.
Sine
Sine was a precolonial Serer kingdom in what is now Senegal, known for its rich Serer culture, religious traditions, and resistance to Islamic and later French expansion.
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C.
Trigonometry and Double Algebra
"Trigonometry and Double Algebra" is a 19th-century mathematical treatise by Augustus De Morgan that develops trigonometric theory alongside an early systematic treatment of complex numbers and symbolic algebra.
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D.
Trigon
Trigon is a powerful demonic supervillain in the Teen Titans universe, best known as Raven’s tyrannical father and a major cosmic threat.
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E.
Thales’ theorem
Thales’ theorem is a fundamental result in Euclidean geometry stating that any angle inscribed in a semicircle is a right angle.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pythagorean identity in trigonometry Target entity description: The Pythagorean identity in trigonometry is a fundamental relation among sine and cosine functions, stating that for any angle θ, sin²θ + cos²θ = 1.
-
A.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
-
B.
Sine
Sine was a precolonial Serer kingdom in what is now Senegal, known for its rich Serer culture, religious traditions, and resistance to Islamic and later French expansion.
-
C.
Trigonometry and Double Algebra
"Trigonometry and Double Algebra" is a 19th-century mathematical treatise by Augustus De Morgan that develops trigonometric theory alongside an early systematic treatment of complex numbers and symbolic algebra.
-
D.
Trigon
Trigon is a powerful demonic supervillain in the Teen Titans universe, best known as Raven’s tyrannical father and a major cosmic threat.
-
E.
Thales’ theorem
Thales’ theorem is a fundamental result in Euclidean geometry stating that any angle inscribed in a semicircle is a right angle.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | trigonometric identity ⓘ |
| alsoKnownAs |
Pythagorean trigonometric identity
ⓘ
linked to:
Pythagorean identity in trigonometry
fundamental trigonometric identity ⓘ |
| basedOn | unit circle definition of sine and cosine ⓘ |
| category | elementary trigonometry ⓘ |
| derivedFrom | Pythagorean theorem ⓘ |
| domainOfθ | all real numbers ⓘ |
| equivalentForm |
1 = sin²θ + cos²θ
ⓘ
cos²θ = 1 − sin²θ ⓘ sin²θ = 1 − cos²θ ⓘ |
| field |
mathematics
ⓘ
trigonometry ⓘ |
| generalizationOf | x² + y² = 1 for points (x,y) on the unit circle ⓘ |
| geometricInterpretation | relationship between legs and hypotenuse of a right triangle on the unit circle ⓘ |
| historicalOrigin | rooted in ancient Greek geometry ⓘ |
| holdsFor |
all complex angles θ
ⓘ
all real angles θ ⓘ |
| implies |
0 ≤ cos²θ ≤ 1
ⓘ
0 ≤ sin²θ ≤ 1 ⓘ |
| nameOrigin | named after the Pythagorean theorem ⓘ |
| prerequisiteFor |
Fourier analysis
ⓘ
complex analysis of trigonometric functions ⓘ signal processing ⓘ |
| property |
holds identically for all θ in its domain
ⓘ
is an even function relation in θ ⓘ |
| relatedIdentity |
1 + cot²θ = csc²θ
ⓘ
1 + tan²θ = sec²θ ⓘ |
| statement | sin²θ + cos²θ = 1 ⓘ |
| symmetry | invariant under θ → −θ ⓘ |
| trueFor |
θ = 0
ⓘ
θ = 2π ⓘ θ = π ⓘ θ = π/2 ⓘ |
| usedFor |
converting between sine and cosine
ⓘ
proving other trigonometric identities ⓘ simplifying trigonometric expressions ⓘ solving trigonometric equations ⓘ |
| usedIn |
calculus
ⓘ
engineering ⓘ geometry ⓘ physics ⓘ |
| usesFunction |
cosine
ⓘ
sine ⓘ |
| variable | θ ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pythagorean identity in trigonometry Description of subject: The Pythagorean identity in trigonometry is a fundamental relation among sine and cosine functions, stating that for any angle θ, sin²θ + cos²θ = 1.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pythagorean identity in trigonometry
linked to: Pythagorean identity in trigonometry