AoPSWiki
Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

Trigonometric substitution

From AoPSWiki

Trigonometric substitution is the technique of replacing variables in equations with \sin \theta\, or \cos {\theta}\, or other functions from trigonometry.

In calculus, it is used to evaluate integrals of expressions such as \sqrt{a^2+x^2},\sqrt{a^2-x^2} or \sqrt{x^2-a^2}

Contents

Examples

\sqrt{a^2+x^2}

To evaluate an expression such as \int \sqrt{a^2+x^2}\,dx, we make use of the identity \tan^2x+1=\sec^2x. Set x=a\tan\theta and the radical will go away.


\sqrt{a^2-x^2}

Making use of the identity \displaystyle\sin^2\theta+\cos^2\theta=1, simply let x=a\sin\theta.


\sqrt{x^2-a^2}

Since \displaystyle\sec^2(\theta)-1=\tan^2(\theta), let x=a\sec\theta.



This article is a stub. Help us out by expanding it.

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Trigonometric_substitution"
Want to learn how to tackle those tough AMC/AIME/Olympiad counting and probability problems? Check out Art of Problem Solving's Intermediate Counting & Probability by David Patrick.
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us