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

so hmmm which is it, the +/-?   well, we know that tan(θ) < 0, that means the tangent of the angle is negative, well, the tangent is opposite/adjacent, the only way that fraction can be negative, is that if either, no both, just either opposite or adjacent is negative.

well, we know the opposite is +1, so, the adjacent then has to be negative, so is -2√(2) then

The volume is 56.8 triangular the