Weierstrass Substitution
Weierstrass substitution t = tan(x/2) converts trig integrals to rational functions.
Step-by-step examples and derivation.
The Formula
sinθ = 2t / (1 + t2)
cosθ = (1 − t2) / (1 + t2)
dθ = 2 / (1 + t2) dt
The Weierstrass substitution, also called the tangent half-angle substitution, is a technique that converts trigonometric integrals into rational function integrals. By setting t = tan(θ/2), every trigonometric function can be expressed as a rational function of t. This is powerful because rational functions can always be integrated using partial fractions.
The substitution is named after Karl Weierstrass, the German mathematician who formalized many concepts in analysis during the 19th century. However, the technique was known to earlier mathematicians including Euler. It is sometimes called the universal trigonometric substitution because it works on any integral involving trigonometric functions, unlike other methods that only work in specific cases.
The derivation starts from the half-angle identities. Since t = tan(θ/2), we can construct a right triangle where the opposite side is t and the adjacent side is 1, giving a hypotenuse of √(1 + t²). From this triangle, sin(θ/2) = t/√(1 + t²) and cos(θ/2) = 1/√(1 + t²). Applying double angle formulas then yields the expressions for sinθ and cosθ in terms of t.
While the Weierstrass substitution always works, it can sometimes lead to unnecessarily complicated integrals. Experienced mathematicians first check whether simpler methods (like direct substitution or symmetry arguments) apply before resorting to this universal technique.
Variables
| Symbol | Meaning |
|---|---|
| t | The substitution variable, equal to tan(θ/2) |
| θ | The original angle variable |
| dθ | The differential, transformed to dt |
Example 1
Express sin(60°) using the Weierstrass substitution.
t = tan(30°) = 1/√3 ≈ 0.5774
sin(60°) = 2t / (1 + t²) = 2(0.5774) / (1 + 0.3333) = 1.1547 / 1.3333
sin(60°) ≈ 0.8660 = √3/2 ✔
Example 2
Convert ∫ dθ / (1 + sinθ) using the substitution.
Replace sinθ = 2t/(1+t²) and dθ = 2dt/(1+t²)
∫ [2/(1+t²)] / [1 + 2t/(1+t²)] dt = ∫ 2/(1+t²+2t) dt = ∫ 2/(1+t)² dt
= −2/(1+t) + C = −2/(1 + tan(θ/2)) + C
When to Use It
The Weierstrass substitution is a universal tool for trigonometric integration.
- Integrating rational functions of sine and cosine
- Solving integrals that resist simpler substitution methods
- Converting trigonometric equations into polynomial equations
- Theoretical proofs in analysis requiring rational parameterization of the unit circle
- Engineering applications involving periodic function integration