Final answer:
The angle θ, with a negative cosine and negative sine values, must be in the third quadrant (Quadrant III) where both x and y coordinates are negative.
Step-by-step explanation:
Given that cosθ = -8/17 and sinθ is negative, if we want to determine in which quadrant angle θ lies, we need to recall that the sign of the cosine and sine functions corresponds to the x and y coordinates in the Cartesian plane, respectively. Since the cosine is negative and sine is also negative, θ must be in the third quadrant (Quadrant III), where both x and y coordinates are negative. This can be derived from the concept that in the first quadrant (Quadrant I), both sine and cosine are positive, in the second quadrant (Quadrant II) cosine is negative and sine is positive, in the third quadrant (Quadrant III) both are negative, and in the fourth quadrant (Quadrant IV) cosine is positive and sine is negative.