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.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | trigonometric identity ⓘ |
| alsoKnownAs |
Pythagorean trigonometric identity
NERFINISHED
ⓘ
fundamental trigonometric identity ⓘ |
| basedOn | unit circle definition of sine and cosine ⓘ |
| category | elementary trigonometry ⓘ |
| derivedFrom | Pythagorean theorem NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.