Use a half measure identity to find the exact value..
Cos t= 1/3 and 0 <t<1/2 pie find sin 1/2 t..
use cos 2A = 1 - sin^2 A
or
cos t = 1 - 2sin^2 (t/2)
2sin^2 (t/2) = 1 - 1/3 = 2/3
sin^2 (t/2) = 1/3
sin (t/2) = 1/√3 , since you want quadrant I, we use only the positive result