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Chapter 5: Problem 73

Use the trigonometric substitution \(u=a \tan \theta,\) where \(-\pi / 2<\theta<\pi / 2\) and \(a>0,\) to simplify the expression \(\sqrt{a^{2}+u^{2}}\)

Short Answer

Expert verified
The simplified expression for \(\sqrt{a^{2}+u^{2}}\), using the trigonometric substitution \(u=a \tan \theta\), is \(a\sec \theta\).

Step by step solution

01

Substitute

Replace \(u\) in the expression \(\sqrt{a^{2}+u^{2}}\) with the given substitution, \(u=a \tan \theta\). That results in \(\sqrt{a^{2}+a^{2} \tan^{2} \theta}\).
02

Factor Out

Now, factor out \(a^{2}\) from under the square root, this will result in \(\sqrt{a^{2}(1+\tan^{2} \theta)}\).
03

Use Trigonometric Identity

Notice that \(1+\tan^{2} \theta\) is actually the trigonometric identity for \(\sec^{2} \theta\). Replace \(1+\tan^{2} \theta\) with \(\sec^{2} \theta\). That simplifies the expression to \(\sqrt{a^{2}\sec^{2}\theta}\).
04

Simplify the Square Root

Finally, we can simplify the expression inside the square root to gives us the final answer. The square and square root cancel each other out, yielding \(a\sec \theta\).

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