Assuming you are using Mathematica
Reduce[{P1 == x + y Sin[2theta1] + z Cos[2theta1],
P2 == x + y Sin[2theta2] + z Cos[2theta2],
P3 == x + y Sin[2theta3] + z Cos[2theta3]},{x,y,z}]
The result is many different possibilities
(condition && x==expression && y==expression && z==expression)||
(condition && x==expression && y==expression && z==expression)||
...
The solution may be much simpler if you insert the values for P and theta first
