How can I obtain a solution for this system of equations?

Questions
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1

I am trying to solve this system of three equations with three variables, but I am doing something wrong. This is the system:
P1 == x + ysin (2theta1) + zcos (2theta1),
P2 == x + ysin (2theta2) + zcos (2theta2),
P3 == x + ysin (2theta3) + zcos (2theta3)},
where the unknown variables are {x, y, z}] and I want to obtain an equation for each one of these variables depending on the P and theta which are values that I will know.

thank you for your help

2

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

3

I would rather use Solve, for a much more readable expression:

Solve[{P1 == x + y Sin[2 theta1] + z Cos[2 theta1],
   P2 == x + y Sin[2 theta2] + z Cos[2 theta2],
   P3 == x + y Sin[2 theta3] + z Cos[2 theta3]},
  {x, y, z}] // FullSimplify