Obviously, the answer will be negative because cosine is negative in quadrant III. We know two of the sides of the triangle, based on the sine, which is the opposite over the hypotenuse = 12/13. To find the third side, which would be the adjacent side, use the Pythagorean Theorem:
a² + b² = c²
Let a = adjacent side
We know b and c; b is the opposite side and c is the hypotenuse. Once you find the adjacent side, you can compute the cosine of the angle because cosine is the adjacent side divided by the hypotenuse.
Can you take it from here?