If cosθ = −3/5 and θ lies in the third quadrant, then tanθ equals:

si$n^2$θ = 1 − 9/25 = 16/25, and sine is negative in the third quadrant, so sinθ = −4/5. Then tanθ = $(-4/5)/(-3/5)=4/3$. Tangent is positive in the third quadrant because both sine and cosine are negative there — taking only the cosine's sign into account produces −4/3.