Weierstrass substitution formulas

The Weierstrass substitution formulas for $-\\pi<x<\\pi$ are:

$\\begin{array}[]{rl}\\sin x&=\\displaystyle\\frac{2t}{1+t^{2}}\\\\&\\\\\\cos x&=\\displaystyle\\frac{1-t^{2}}{1+t^{2}}\\\\&\\\\dx&=\\displaystyle\\frac{2}{1+t^{2}}\\,dt\\end{array}$

They can be obtained in the following manner:

Make the Weierstrass substitution $\\displaystyle t=\\tan\\left(\\frac{x}{2}\\right)$ . (This substitution is also known as the universal trigonometric substitution.) Then we have

$\\begin{array}[]{rl}\\displaystyle\\cos\\left(\\frac{x}{2}\\right)&=\\displaystyle%\\frac{1}{\\displaystyle\\sec\\left(\\frac{x}{2}\\right)}\\\\&\\\\&=\\displaystyle\\frac{1}{\\displaystyle\\sqrt{1+\\tan^{2}\\left(\\frac{x}{2}\\right)}%}\\\\&\\\\&=\\displaystyle\\frac{1}{\\sqrt{1+t^{2}}}\\end{array}$

and

$\\begin{array}[]{rl}\\displaystyle\\sin\\left(\\frac{x}{2}\\right)&=\\displaystyle%\\cos\\left(\\frac{x}{2}\\right)\\cdot\\tan\\left(\\frac{x}{2}\\right)\\\\&\\\\&=\\displaystyle\\frac{t}{\\sqrt{1+t^{2}}}.\\end{array}$

Note that these are just the “formulas involving radicals (http://planetmath.org/Radical6)” as designated in the entry goniometric formulas; however, due to the restriction on $x$ , the $\\pm$ ’s are unnecessary.

Using the above formulas along with the double angle formulas, we obtain

$\\begin{array}[]{rl}\\sin x&=\\displaystyle 2\\sin\\left(\\frac{x}{2}\\right)\\cdot%\\cos\\left(\\frac{x}{2}\\right)\\\\&\\\\&=\\displaystyle 2\\cdot\\frac{t}{\\sqrt{1+t^{2}}}\\cdot\\frac{1}{\\sqrt{1+t^{2}}}\\\\&\\\\&=\\displaystyle\\frac{2t}{1+t^{2}}\\end{array}$

and

$\\begin{array}[]{rl}\\cos x&=\\displaystyle\\cos^{2}\\left(\\frac{x}{2}\\right)-\\sin^%{2}\\left(\\frac{x}{2}\\right)\\\\&\\\\&=\\displaystyle\\left(\\frac{1}{\\sqrt{1+t^{2}}}\\right)^{2}-\\left(\\frac{t}{\\sqrt{%1+t^{2}}}\\right)^{2}\\\\&\\\\&=\\displaystyle\\frac{1}{1+t^{2}}-\\frac{t^{2}}{1+t^{2}}\\\\&\\\\&=\\displaystyle\\frac{1-t^{2}}{1+t^{2}}.\\end{array}$

Finally, since $\\displaystyle t=\\tan\\left(\\frac{x}{2}\\right)$ , solving for $x$ yields that $x=2\\arctan t$ . Thus, $\\displaystyle dx=\\frac{2}{1+t^{2}}\\,dt$ .

The Weierstrass substitution formulas are most useful for integrating rational functions of sine and cosine (http://planetmath.org/IntegrationOfRationalFunctionOfSineAndCosine).

Title	Weierstrass substitution formulas
Canonical name	WeierstrassSubstitutionFormulas
Date of creation	2013-03-22 17:05:25
Last modified on	2013-03-22 17:05:25
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	12
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 26A36
Classification	msc 33B10
Synonym	Weierstraß substitution formulas
Related topic	GoniometricFormulae
Related topic	IntegrationOfRationalFunctionOfSineAndCosine
Related topic	PolynomialAnalogonForFermatsLastTheorem
Defines	Weierstrass substitution
Defines	Weierstaß substitution
Defines	universal trigonometric substitution