Answer:
Quadrant III, sin θ = -12/13
Explanation:
To solve this problem we can look at a unit circle (attached below). We can observe the following about the quadrants:
cos θ = 5/13 falls into the first OR fourth quadrant since cos (67) = 5/13 (roughly, obtained with inverse cosine function). However, we also have the restriction that sin θ < 0, therefore the quadrant that cos θ falls into is the fourth quadrant.
To find sin θ we first consider that cosine is adjacent over hypoteneuse. So, a = 5 and h = 13. Using pythagorean theorem we find that o (opposite) is equal to 12. Since sine is opposite over hypoteneuse we have:
sin θ = ±12/13
We use the plus or minuse sign because sin θ maps to both a positive and negative value, even though we used a geometrical approach, we still have to consider this. Of course, we know that sin θ has to be negative (given in the problem) so sin θ is -12/13.