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.
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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| implies |
dx = 2 dt / (1 + t^2)
ⓘ
x = 2 arctan(t) ⓘ |
| limitation | can lead to algebraically complicated expressions ⓘ |
| originalVariable | x ⓘ |
| relatedTo |
Pythagorean identity
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.