Remember too that the current and voltage will not be 90 degrees out of phase due to the inherent resistance of the coil. If the resistance is substantial compared to the inductive reactance, then the phase shift may well be much smaller.
Another way to measure the phase shift is to switch your 'scope to X-Y mode and use Lissajous patterns.
If you use a small resistance at the "low" end of the circuit as Joe suggests, you can feed the voltage from that (which represents current) to one input. Connect the other input directly to the supply, and adjust X and Y gain for equal height and width of the display.
The shape now indicates the phase shift, in phase being represented by a straight line at 45 degrees, this gradually opening out to an ellipse as the phase angle increases and becoming a circle at 90 degrees, as on the top line here:
You can calculate the actual phase shift by centering the display and measuring the ratio of the distance between the points at which the trace crosses the zero line and overall size, like this:
Xzero / Xtotal is then equal to the sine of the phase angle (you need to adjust for which quadrant you're in -- the display above is an angle between 90 and 180 degrees).
[This message has been edited by pauluk (edited 07-05-2006).]