Difference between revisions of "Trigonometric substitution"

Revision as of 23:39, 7 July 2006

Trigonometric substitution is the technique of replacing unknown variables in equations with $\sin \theta\,$ or $\cos {\theta}\,$

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

and

$\sqrt{a^2+x^2}$

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 $x=a\sec\theta$ .

Examples

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

@@ Line 1: / Line 1: @@
 Trigonometric substitution is the technique of replacing unknown variables in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math>
-It frequently relies on using [[trigonometric identities]].
+In calculus, it is used to evaluate integrals with expressions such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math>
+ and <math>\sqrt{x^2-a^2}</math>
-Trigonometric Substitution is also a Calculus method to compute integrals.
+== <math>\sqrt{a^2+x^2}</math> ==
+To evaluate an expression such as <math>\int \sqrt{a^2+x^2}\,dx</math>, we make use of the identity <math>\tan^2x+1=\sec^2x</math>.  Set <math>x=a\tan\theta</math> and the radical will go away.
- This article is incomplete (did I need to tell you that?).
+== <math>\sqrt{a^2-x^2}</math> ==
+Making use of the identity <math>\displaystyle\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>.
+== <math>\sqrt{x^2-a^2}</math> ==
+Since <math>\displaystyle\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>.
+== Examples ==
+{{stub}}

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