sin(θ) < 0 is another way to say sin(θ) is negative.
tan(θ) < 0 is another way to say tan(θ) is negative.
let's recall that, on the III Quadrant sine and cosine are both negative, and thus the tangent is positive, recall tan(θ)=sin(θ)/cos(θ).
on the IV Quadrant however, sine is negative and cosine is positive, thus tangent is negative.
