If ##sin theta = -1/3##, and ##180° < theta < 270°##, what is ##tan theta##?

##tantheta = 1/sqrt(8) = sqrt(8)/8##

Because theta is greater than 180 and less than 270, you know this is a 3rd quadrant triangle. So I have included a 3rd quadrant triangle. They tell you that ##sintheta = -1/3##

Because ##sintheta## is equal to opposite/hypotenuse you can draw a triangle as follows:

Now, use to find the adjacent side: ##(-1)^2 + (adjacent)^2 = 3^2## ##1 + (adjacent)^2 = 9## ##(adjacent)^2 = 8## ##sqrt(adjacent^2) = +-sqrt(8)## Because this is a 3rd quadrant triangle we choose the negative root: ##adjacent = -sqrt(8)##

We're almost done. ##tantheta = (opposite)/(adjacent)## Thus, ##tantheta = -1/-sqrt(8) = 1/sqrt(8) = sqrt(8)/8##