For the most part, will a law of cosines always be one triangle? As in one triangle to solve?

Answer 1

yes. The reason the law of sines can give two triangles is because sin(x) is positive all the way from 0 to 180.

cos(x) becomes negative for x>90, so the formula takes that into account, always leaving only one possible answer.

I mean, think about it geometrically. If you know the lengths of two sides, and the angle between them, there's only one line segment that connects the two endpoints.