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.