Question

Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that $\mathbf{0}\mathbf{\text{<}}\mathit{\theta }\mathbf{\text{<}}\frac{\mathbf{\pi }}{\mathbf{2}}$$\sqrt{\mathbf{9}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{,}\mathit{x}\mathbf{=}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$

Short answer

The simplified expression is$\sqrt{\mathbf{9}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\mathbf{=}\mathbf{3}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$

Step-by-step solution

Step 1. Given information

An expression $\sqrt{9-x^2}$ and value for substitution as$x=3\sin \theta$

Step 2. Concept used

First substitute the value of x in the expression given in question. After that, with the help of trigonometric identities simplify it into simplest form. The simplest form is the correct answer.

Step 3. Calculation

Now, substitute the value in$\sqrt{9-x^2}$ and simplify:

$\begin{array}{l}\sqrt{9-x^2}=\sqrt{9-{\left(3\sin \theta \right)}^2}\\ =\sqrt{9-9{\sin }^2\theta }\\ =\sqrt{9\left(1-{\sin }^2\theta \right)}\\ \because {\sin }^2\theta +{\cos }^2\theta =1\\ =\sqrt{9\left({\cos }^2\theta \right)}\\ =3\cos \theta \end{array}$

After simplification, we get-

$3\cos \theta$