tangent half-angle substitution
E480875
Tangent half-angle substitution is a trigonometric technique that converts integrals involving sine and cosine into rational functions by substituting in terms of the tangent of half the angle.
All labels observed (1)
| Label | Occurrences |
|---|---|
| tangent half-angle substitution canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4927360 — 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: tangent half-angle substitution Context triple: [Weierstrass substitution, alsoKnownAs, tangent half-angle substitution]
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A.
Radau
Radau is a river in Lower Saxony, Germany, known as a tributary of the Oker and for flowing through the Harz region.
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B.
Seča
Seča is a small coastal settlement in southwestern Slovenia, situated near the town of Piran on the Adriatic Sea.
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C.
Tangen
Tangen is a former settlement in Norway that was incorporated into the city of Drammen through a municipal merger.
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D.
Tangtse
Tangtse is a village in the Leh district of Ladakh, India, situated along key routes between the Indus Valley and the Pangong Tso region in the Himalayas.
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E.
Trizina
Trizina is a modern Greek settlement located near the ancient city of Troezen in the northeastern Peloponnese.
- 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: tangent half-angle substitution Target entity description: Tangent half-angle substitution is a trigonometric technique that converts integrals involving sine and cosine into rational functions by substituting in terms of the tangent of half the angle.
-
A.
Radau
Radau is a river in Lower Saxony, Germany, known as a tributary of the Oker and for flowing through the Harz region.
-
B.
Seča
Seča is a small coastal settlement in southwestern Slovenia, situated near the town of Piran on the Adriatic Sea.
-
C.
Tangen
Tangen is a former settlement in Norway that was incorporated into the city of Drammen through a municipal merger.
-
D.
Tangtse
Tangtse is a village in the Leh district of Ladakh, India, situated along key routes between the Indus Valley and the Pangong Tso region in the Himalayas.
-
E.
Trizina
Trizina is a modern Greek settlement located near the ancient city of Troezen in the northeastern Peloponnese.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
calculus technique
ⓘ
integration technique ⓘ mathematical method ⓘ trigonometric substitution method ⓘ |
| advantage |
provides a systematic method for many trigonometric integrals
ⓘ
reduces trigonometric integrals to algebraic integrals ⓘ |
| alsoKnownAs |
Weierstrass substitution
ⓘ
universal trigonometric substitution ⓘ |
| appliesTo |
definite integrals
ⓘ
indefinite integrals ⓘ |
| basedOn | substitution t = tan(x/2) ⓘ |
| category |
methods of integration
ⓘ
trigonometric transformations ⓘ |
| converts | integrals of rational functions of sin(x) and cos(x) into integrals of rational functions of t ⓘ |
| defines | t = tan(x/2) ⓘ |
| domainCondition | x not equal to odd multiples of π ⓘ |
| expresses |
1 + cos(x) = 2 / (1 + t^2)
ⓘ
1 - cos(x) = 2t^2 / (1 + t^2) ⓘ cos(x) = (1 - t^2) / (1 + t^2) ⓘ cos^2(x) = (1 - t^2)^2 / (1 + t^2)^2 ⓘ sin(x) = 2t / (1 + t^2) ⓘ sin^2(x) = 4t^2 / (1 + t^2)^2 ⓘ tan(x) = 2t / (1 - t^2) ⓘ |
| field |
calculus
ⓘ
integral calculus ⓘ trigonometry ⓘ |
| historicalAttribution | Karl Weierstrass ⓘ |
| implies |
dx = 2 dt / (1 + t^2)
ⓘ
x = 2 arctan(t) ⓘ |
| limitation | can lead to algebraically complicated expressions ⓘ |
| originalVariable | x ⓘ |
| relatedTo |
Pythagorean identity
ⓘ
linked to:
Pythagorean identity in trigonometry
double-angle formulas ⓘ half-angle formulas ⓘ trigonometric identities ⓘ |
| requires | algebraic manipulation of rational functions ⓘ |
| solves | integrals of the form ∫R(sin x, cos x) dx where R is rational ⓘ |
| step |
back-substitute t = tan(x/2) to express result in x
ⓘ
integrate resulting rational function in t ⓘ replace dx with 2 dt / (1 + t^2) ⓘ substitute trigonometric functions in terms of t ⓘ |
| typicalVariable | t ⓘ |
| usedFor |
converting trigonometric integrals into rational functions
ⓘ
evaluating integrals involving sine and cosine ⓘ simplifying integrals of rational functions of sine and cosine ⓘ |
| usedIn |
integration techniques in analysis
ⓘ
university-level calculus courses ⓘ |
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: tangent half-angle substitution Description of subject: Tangent half-angle substitution is a trigonometric technique that converts integrals involving sine and cosine into rational functions by substituting in terms of the tangent of half the angle.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.