How to find rotation angle given output of cos?

I have a unit quaternion q to represent a rotation around an axis. q = (cos(rotation-angle/2), unit vector of rotation axis)

My q has come out to be (-1/sqrt(6), -1/sqrt(6)i, 1/sqrt(2)j, -1/sqrt(6)k).

I don't know how to find the angle passed into cosine given that the output of cosine is -1/sqrt(6).

How would I use arccos to solve this? If I only had x and y I would think to just use the i or j from the vector as the opposite or adjacent...but how do I extend that to work with 1/sqrt(6) in the z direction?

Is there a different and/or easier way than using arccos?

Thanks for any help.