If sin theta = 1/3 and tan theta < 0, what is the value of cos theta?

a) 2 square root 2/3
b) -2 square root/3
c) 2 square root 2
d) -2 square root 2

If sin Ф=1/3>0 and tan Ф<0; then Ф belongs to the second quadrant and cos Ф will be negative (cos Ф<0).

sin²Ф+cos²Ф=1
(1/3)²+cos²Ф=1
1/9+cos²Ф=1
cos²Ф=1-1/9
cos²Ф=8/9
cos Ф=⁺₋√(8/9)=⁺₋2√2 / 3

We have two possible solutions.
cos Ф=2√2 / 3 This solutions is not possible, because in this case cos Ф has to be negative (cos Ф<0)

cos Ф=(-2√2) / 3 this solutions is right.

B. -2 square root 2 /3    (or    (-2√2) /3  )