Question

Couldn't you use the law of cosines to find one angle and use the law of sines to find the rest

Answer 1

Correct. In the case where you are given enough information to use the law of cosines you could in fact then use the law of sines afterwards to find your remaining angle. That being said beware of solutions that don't make a feasible triangle (if you were using the law of cosines you only have one angle, so that means whatever your second angle is that you found using the law of sines can't make your sum go over 180, because you still need some angle left for the last angle).