Weierstrass Substitution/Cosine
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Proof Technique
Let:
- $u \leftrightarrow \tan \dfrac \theta 2$
for $-\pi < \theta < \pi$, $u \in \R$.
Then:
- $\cos \theta = \dfrac {1 - u^2} {1 + u^2}$
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:
\(\ds u\) | \(=\) | \(\ds \tan \dfrac \theta 2\) | ||||||||||||
\(\ds \cos \theta\) | \(=\) | \(\ds \dfrac {1 - u^2} {1 + u^2}\) | Tangent Half-Angle Substitution for Cosine |
$\blacksquare$
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
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.