Did you mean cos(x) < 0 instead of arccos(x) < 0?
Actually I have suspected that you are confused as to the difference in those two functions.
Here is an example. Two lines where .
Then the angle between is .
Now if then is acute.
And if then is obtuse.
But if your question were to ask for the acute angle betwwen then it is .
If I remember correctly, I got a positive angle, alpha, when taking the inverse cosine and then did pi - alpha = theta, where cos(theta) gave me the correct answer.
Is there something wrong with what I just said (in the post I am writing right now)?