Specifically speaking, 3-SAT is an example of NP-Complete problem. But your problem is only considered with NP problems.
We have the decision problem:
Is a SAT problem in NP?
Obviously, it is. Because you could always verify a "certificate" in time linear to your input size (values of your literals, that is a sequences of 0s and 1s).
For your question:
Is a un-SAT problem in co-NP? (complement to SAT problem)
We could find the following definition:
A decision problem X is a member of co-NP if and only if its
complement X is in the complexity class NP .
Therefore, un-SAT is co-NP by definition.