Questions: Use the suggested substitution to write the expression as a trigonometric expression. Simplify your answer as much as possible. Assume (0 leq theta leq fracpi2). [ sqrt81-9 x^2 cdot fracx3=sin (theta) ]

$Use the suggested substitution to write the expression as a trigonometric expression. Simplify your answer as much as possible. Assume (0 leq theta leq fracpi2). [ sqrt81-9 x^2 cdot fracx3=sin (theta) ]$

Transcript text: Use the suggested substitution to write the expression as a trigonometric expression. Simplify your answer as much as possible. Assume $0 \leq \theta \leq \frac{\pi}{2}$. \[ \sqrt{81-9 x^{2}} \cdot \frac{x}{3}=\sin (\theta) \]

Solution

Solution Steps

Step 1: Define the Expression

We start with the expression given in the problem: \[ \sqrt{81 - 9x^2} \cdot \frac{x}{3} \]

Step 2: Apply the Substitution

We use the substitution $ x = 3\sin(\theta) $. Substituting this into the expression, we have: \[ \sqrt{81 - 9(3\sin(\theta))^2} \cdot \frac{3\sin(\theta)}{3} \]

Step 3: Simplify the Expression

Simplifying inside the square root: \[ 81 - 9(9\sin^2(\theta)) = 81 - 81\sin^2(\theta) \] This simplifies to: \[ \sqrt{81(1 - \sin^2(\theta))} \] Using the Pythagorean identity $1 - \sin^2(\theta) = \cos^2(\theta)$, we have: \[ \sqrt{81\cos^2(\theta)} = 9\cos(\theta) \]

Step 4: Combine and Simplify Further

The expression now becomes: \[ 9\cos(\theta) \cdot \sin(\theta) \] This simplifies to: \[ 9\sin(\theta)\cos(\theta) \]