I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range $0-\pi$. I need a result in the range $\pi-2\pi$.

Let's say that $\vec x$ is a vector in the positive x-direction and $\vec y$ is a vector in the positive y-direction and $\vec z$ is a reference vector in the positive z-direction. $\vec z$ is perpendicular to both $\vec x$ and $\vec y$. Would it then be possible to calculate the angle between $\vec x$ and $\vec y$ and get a result in the range $\pi-2\pi$? 

The angle value should be measured counter clockwise. I have not been able to figure out how to do that.
I am no math guru but I have basic understanding of vectors at least.
Thank you very much for the help!