Question

suppose that sin0=-5/13 and that 0 is a quadrant III angle. what is the exact value of cos0

Answer 1

The "reference angle" that theta makes with the -x axis is sin^-1(5/13) . The cosine of that angle is 12/13.
(Think of a 5:12:13 right triangle)

Cosine is negative in the third quadrant.

The answer is -12/13

Answer 2

make a sketch of the right-angled triangle in quad III, with hypotenuse 13 and opposite 5.
By Pythagoras the adjacent would be 12 (the 5-12-13 triangle)
so cosØ = -12/13 ( by CAST, in III the cosine is negative)