Verified answer

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

so cos(x) = +/- √(1-sin^2(x))

Since this is in the second quadrant: - √(1-sin^2(x))

You need to know what sin(theta) is.

Lets say sin(theta) =x

Draw a triangle and label an angle as theta, the opposite side x the hypothenuse is 1 and adjacent sqrt(1-x^2) from Pythagorean theorem. The hyp is 1 because x is written as x/1 and sin is opp/hyp

So cos is negative in second quadrant and is adj/hyp

So cos(theta) = -sqrt(1-x^2)... Now take your value of sin(theta) and plug it into x.

Start with (sin(θ))^2 + (cos(θ))^2 =1.

Solving for cos(θ) gives two versions; cos(θ) = sqrt( 1- (sin(θ))^2 ) or cos(θ) = -sqrt( 1- (sin(θ))^2 )

use the quadrant (π/2<θ<π) to determine which values are appropriate pos or neg.