Use the suggested substitution to write the expression as a trigonometric expression. Simplify your answer as much as possible. Assume 0 ≤ θ ≤ π2. ?

\sqrt(100-4x^(2)),(x)/(5)=sin(\theta )

1 Answer
Feb 22, 2018

sqrt(100-4x^2) = 10cos(theta) where x/5 = sin(theta).

Explanation:

sqrt(100-4x^2)

x/5 = sin(theta)

We may first rewrite the second equation in terms of x in order to substitute it into the first.

color(red)x = color(red)(5sin(theta))

sqrt(100-4color(red)x^2)

= sqrt(100 - 4(5sin(theta))^2)

= sqrt(100 - 4(25sin^2(theta)))

= sqrt(100 - 100sin^2(theta))

Conveniently, we are able to factor out 100 and simplify the radical.

= sqrt(100*(1 - sin^2(theta)))

= sqrt(100)*sqrt(1 - sin^2(theta))

= 10color(blue)(sqrt(1 - sin^2(theta)))

By the Pythagorean identity:

sin^2(theta) + cos^2(theta) = 1

cos^2(theta) = 1 - sin^2(theta)

color(blue)(cos(theta)) = color(blue)(sqrt(1 - sin^2(theta)))

Therefore,

10color(blue)(sqrt(1 - sin^2(theta)))

= 10cos(theta)