Coil in magnetic field. Voltage step response
The physical process

The voltage induced by a changing magnetic field is given by Faraday's law. Basically it says 

voltage = turns of wire * (rate of change of magnetic flux)

Where magnetic flux basically means the total magnetic field passing through the coil.

Your example

If your coil is stationary, the rate of change of magnetic flux due to the magnet is zero. So, in the experiment your propose, the angle will have no effect. You won't be able to sense the magnetic field angle using the coil alone. 

That said, you might be able to sense the presence of iron near the coil, because the inductance of the coil will be larger if iron is nearby (which will manifest as a slower rise and slower fall in the current).