Now that I've got the first value, how do I find the second? All I know is that the CAST rule may be involved, but I don't know how to apply it in this situation. Help on this would be greatly appreciated.
Actually it's because cos(x) is an even function, meaning cos(x) = cos(-x). It just so happens that cos(-x) = cos(360 - x). I figured you were probably looking for an answer in the range [0, 360], so I gave the latter answer.
The corresponding identity for sin(x) is sin(x) = -sin(-x) because sin(x) is odd.
It is also true that sin(x) = sin(180-x) but this more closely corresponds with cos(x) = -cos(180-x). See here.