If x = 5 sin θ, use trigonometric substitution to write √25-x^2 as a trigonometric function of θ, where -∏/2 < θ < ∏/2

Question

If x = 5 sin θ, use trigonometric substitution to write √25-x^2 as a trigonometric function of θ, where -∏/2 < θ < ∏/2

Answer 1

Given:

x = 5 sin θ

We are asked to rewrite √(25 - x²) as a function of θ. Simply substitute x and then simplify:

∴√(25 - x²) = √(25 - (5 sin θ)²)

=√(25 - 25 sin²θ)

=√(25 - 25 sin²θ)

Factor out 25 then put it outside the radicand:

=5√(1 - sin²θ)

By Pythagorean identity cos²θ=1 - sin²θ:

=5√(cos²θ)

=5 cos θ