Question

Use the trigonometric substitution to select the algebraic expression as a trigonometric function of theta, where 0 < theta < pi/2

sqrt(4-x^2) x=2sintheta

Answer 1

To use trigonometric substitution to select the algebraic expression as a trigonometric function of θ, substitute x=2sinθ into the expression √(4-x^2). Simplify and rewrite the expression using trigonometric identities, ending with 2cosθ.

Step-by-step explanation:

To use trigonometric substitution to select the algebraic expression as a trigonometric function of θ, we need to substitute x=2sinθ into the expression √(4-x^2). Then, we can simplify and rewrite the expression using trigonometric identities. Let's go through the steps:

Start with the expression √(4-x^2).

Substitute x=2sinθ into the expression: √(4-(2sinθ)^2).

Simplify the expression using the trigonometric identity sin^2θ+cos^2θ=1: √(4-(2sinθ)^2) = √(4-4sin^2θ) = √4(1-sin^2θ) = 2√(1-sin^2θ).

Use the trigonometric identity cos^2θ = 1-sin^2θ to rewrite the expression: 2√(1-sin^2θ) = 2cosθ.