Make the trigonometric substitution x = a sin 𝜃 for −𝜋/2 < 𝜃 < 𝜋/2 and a > 0. Use fundamental identities to simplify the resulting expression.

Question

Make the trigonometric substitution x = a sin 𝜃 for −𝜋/2 < 𝜃 < 𝜋/2 and a > 0.

 Use fundamental identities to simplify the resulting expression.

	a2 − x2
x2

Answer 1

√(a² - x²) = √(a² - a²sin²θ) = √[a²(1 - sin²θ)] = √[a²cos²θ] = lal lcosθl

We know that a > 0 and, since -π/2 < θ < π/2, θ lies in quadrant 1 or 4. So, cosθ > 0.

Therefore, lal lcosθl = acosθ.