# Weierstrass Substitution

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## Proof Technique

The **Weierstrass substitution** is an application of Integration by Substitution.

The substitution is:

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

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

It yields:

\(\ds \sin \theta\) | \(=\) | \(\ds \frac {2 u} {1 + u^2}\) | ||||||||||||

\(\ds \cos \theta\) | \(=\) | \(\ds \frac {1 - u^2} {1 + u^2}\) | ||||||||||||

\(\ds \frac {\d \theta} {\d u}\) | \(=\) | \(\ds \frac 2 {1 + u^2}\) |

This can be stated:

- $\ds \int \map F {\sin \theta, \cos \theta} \rd \theta = 2 \int \map F {\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} } \frac {d u} {1 + u^2}$

where $u = \tan \dfrac \theta 2$.

## 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:

\(\ds u\) | \(=\) | \(\ds \tan \dfrac \theta 2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \theta\) | \(=\) | \(\ds 2 \tan^{-1} u\) | Definition of Inverse Tangent | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d \theta} {\d u}\) | \(=\) | \(\ds \dfrac 2 {1 + u^2}\) | Derivative of Arctangent Function and Derivative of Constant Multiple | ||||||||||

\(\ds \sin \theta\) | \(=\) | \(\ds \dfrac {2 u} {1 + u^2}\) | Tangent Half-Angle Substitution for Sine | |||||||||||

\(\ds \cos \theta\) | \(=\) | \(\ds \dfrac {1 - u^2} {1 + u^2}\) | Tangent Half-Angle Substitution for Cosine |

The result follows from Integration by Substitution.

$\blacksquare$

## Also known as

This technique is also known as **tangent half-angle substitution**.

Some sources refer to these results 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 $1$. - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Important Transformations: $14.58$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**half-angle formulae**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**half-angle formulae**:**1.**

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

- Weisstein, Eric W. "Weierstrass Substitution." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassSubstitution.html