if pi/2 ≤ theta ≤ pi and sin(theta)=5/9, find the exact value of cos(theta) and cot(theta)

Question

if pi/2 ≤ theta ≤ pi and sin(theta)=5/9, find the exact value of cos(theta) and cot(theta)

Answer 1

sin(theta) = 5/9 = opposite side over hypotenuse = o/h, o=5, h=9

adjacent side = a, a^2 = h^2 -o^2 = 9^2-5^2 = 81-25 = 56, a=-sqr56=-2sqr14

cos(theta) = adjacent side over hypotenuse = a/h = -(2sqr14)/9= (-2/9)sqr14

cot(theta) = opposite side over adjacent side = o/a = (5/9)/(-2/9)sqr14 = -5/(2sqr14)