If sin theta = 1/3 and tan theta is less than 0 what is the value of cosine theta

If sin theta is 1/3, then theta is in either QI or Q2.  If tan theta is negative, then theta is in either Q2 or Q4.  Thus, theta is in Q2.

If sin theta = opp / hyp = 1/3, then opp=1 and hyp=3.  Thus, adj=sqrt(3^2-1^2), or sqrt 8, or 2sqrt 2.

Then cos theta = adj / hyp = 2sqrt(2) / 3.

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