Use the suggested substitution to write the expression as a trigonometric expression. Simplify your answer as much as possible. Assume 0≤θ≤π20≤𝜃≤𝜋2.36−4x2⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√36−4𝑥2, x3=sin(θ)

Given the substitution x = 3sin(θ), we can rewrite the expression √(36 - 4x²) as follows:

Step 1: Substitute x = 3sin(θ) into the expression:

√(36 - 4*(3sin(θ))²)

Step 2: Simplify the expression inside the square root:

= √(36 - 4*(9sin²(θ)))

= √(36 - 36sin²(θ))

Step 3: Factor out a 36 from inside the square root:

= √(36(1 - sin²(θ)))

Step 4: Recognize the Pythagorean identity 1 - sin²(θ) = cos²(θ):

= √(36cos²(θ))

Step 5: Take the square root of 36 and separate the square roots:

= 6|cos(θ)|

However, since we are given that 0 ≤ θ ≤ π/2, cos(θ) is positive in this range. Therefore, we can remove the absolute value bars:

= 6cos(θ)

So, the expression √(36 - 4x²) simplifies to 6cos(θ) when x = 3sin(θ).