Weierstrass Substitution/Sine

Proof Technique

Let:

$u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.

Then:

$\sin \theta = \dfrac {2 u} {1 + u^2}$

Proof

Let $u = \tan \dfrac \theta 2$ for $-\pi < \theta < \pi$.

From Shape of Tangent Function, this substitution is valid for all real $u$.

Then:

\(\ds u\)

\(=\)

\(\ds \tan \dfrac \theta 2\)

\(\ds \sin \theta\)

\(=\)

\(\ds \dfrac {2 u} {1 + u^2}\)

Tangent Half-Angle Substitution for Sine

$\blacksquare$

Also known as

The technique of Weierstrass Substitution is also known as Tangent Half-Angle Substitution.

Some sources call these results the Tangent-of-Half-Angle Formulae.

Other sources refer to them merely as the Half-Angle Formulas or Half-Angle Formulae.

Also see

• Hyperbolic Tangent Half-Angle Substitution

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Useful substitutions: Example $1$.

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): half-angle formulae: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): half-angle formulae: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): half-angle formula
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): $t$-formulae
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Tangent-of-half-angle formulae
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Tangent-of-half-angle formulae

• This article incorporates material from Weierstrass substitution formulas on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.