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Difference between revisions of "Trigonometric substitution"
 
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Trigonometric substitution is the technique of replacing unknown variables in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math>
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'''Trigonometric substitution''' is the technique of replacing [[variable]]s in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math> or other [[function]]s from [[trigonometry]].
  
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In calculus, it is used to evaluate integrals with expressions such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math>
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In [[calculus]], it is used to evaluate [[integral]]s of [[expression]]s such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math> or <math>\sqrt{x^2-a^2}</math>
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and <math>\sqrt{x^2-a^2}</math>
 
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== <math>\sqrt{a^2+x^2}</math> ==
 
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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.
 
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== <math>\sqrt{a^2-x^2}</math> ==
 
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Making use of the identity <math>\displaystyle\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>.
 
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== <math>\sqrt{x^2-a^2}</math> ==
 
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Since <math>\displaystyle\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>.
 
  
 
== Examples ==
 
== Examples ==
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=== <math>\sqrt{a^2+x^2}</math> ===
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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. However, the <math>dx</math> will have to be changed in terms of <math>d\theta</math>: <math>dx=a\sec^2\theta</math> <math>d\theta</math>
  
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=== <math>\sqrt{a^2-x^2}</math> ===
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Making use of the identity <math>\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>.
  
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=== <math>\sqrt{x^2-a^2}</math> ===
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Since <math>\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>.
  
 
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[[Category: Trigonometry]]

Latest revision as of 17:42, 30 May 2021

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

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. However, the $dx$ will have to be changed in terms of $d\theta$: $dx=a\sec^2\theta$ $d\theta$

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

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

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

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

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