Rotate the triangle ABP 60 degrees around point B to CBQ, so that AB and CB coincide (P turns to Q).

Connecting PQ

Because BP=BQ, PBQ=60 degrees.

So the triangle BPQ is an equilateral triangle with PQ=PB=4 and the angle BQP=60 degrees.

Because PC=5 and CQ=AP=3.

So the square of CQ+the square of PQ = the square of PC

So CQP angle =90 degrees.

Because angle APB= angle CQP

So the angle APB=60+90= 150 degrees.