Solving a set of 3 Nonlinear EquationsIn the following 3 equations: k 1 cos 2 ⁡ ( θ ) + k 2 sin 2 ⁡ ( θ ) = c 1 2 ( k 2 − k 1 ) cos ⁡ ( θ ) sin ⁡ ( θ ) = c 2 k 1 sin 2 ⁡ ( θ ) + k 2 cos 2 ⁡ ( θ ) = c 3 c 1 , c 2 and c 3 are given, and k 1 , k 2 and θ are the unknowns. What is the best way to solve for the unknowns?

Question

Solving a set of 3 Nonlinear EquationsIn the following 3 equations:      k    1        cos    2    ⁡  (  θ  )  +      k    2        sin    2    ⁡  (  θ  )  =      c    1    2  (      k    2    −      k    1    )  cos  ⁡  (  θ  )  sin  ⁡  (  θ  )  =      c    2        k    1        sin    2    ⁡  (  θ  )  +      k    2        cos    2    ⁡  (  θ  )  =      c    3        c    1  ,       c    2   and       c    3   are given, and       k    1  ,       k    2   and   θ are the unknowns. What is the best way to solve for the unknowns?

isscacabby17 · Accepted Answer

Re-write in terms of double-angle expressions, and define values   A, ,   C:                              k          1                          cos          2                ⁡        θ        +                  k          2                          sin          2                ⁡        θ        =                  c          1                                                    ⟹                        cos        ⁡        2        θ        =                              2                          c              1                        −                          k              1                        −                          k              2                                                          k              1                        −                          k              2                                                  =:        A                            2                  (                      k            1                    −                      k            2                    )                sin        ⁡        θ        cos        ⁡        θ        =                  c          2                                                    ⟹                        sin        ⁡        2        θ        =                              c            2                                              k              1                        −                          k              2                                                  =:        B                                      k          1                          sin          2                ⁡        θ        +                  k          2                          cos          2                ⁡        θ        =                  c          3                                                    ⟹                        cos        ⁡        2        θ        =                                            k              1                        +                          k              2                        −            2                          c              3                                                          k              1                        −                          k              2                                                  =:        C            Now,   A  =  C implies       k    1    +      k    2    =      c    1    +      c    3   (but we&#039;d know that simply by adding your first and third equations together), so that       k    2    =      c    1    +      c    3    −      k    1   and  A  =                    c        1            −              c        3                    2              k        1            −              c        1            −              c        3                  B  =            c      2              2              k        1            −              c        1            −              c        3            Observe that you can already write  tan  ⁡  2  θ  =            c      2                      c        1            −              c        3            which gives you   θ. For       k    1   and       k    2  , note that       A    2    +      B    2    =  1 implies  (      c    1    −      c    3        )    2    +      c    2    2    =            (      2              k        1            −              c        1            −              c        3            )        2  so that                              k          1                                    =                  1          2                          (                      c            1                    +                      c            3                    ±                      (                          c              1                        −                          c              3                                      )              2                        +                          c              2              2                                )                                              k          2                                    =                  1          2                          (                      c            1                    +                      c            3                    ∓                      (                          c              1                        −                          c              3                                      )              2                        +                          c              2              2                                )

Ciara Mcdaniel · Accepted Answer

Hint:      k    1        cos    2    ⁡  (  θ  )  +      k    2        sin    2    ⁡  (  θ  )  =      c    1        k    1        sin    2    ⁡  (  θ  )  +      k    2        cos    2    ⁡  (  θ  )  =      c    3        c    1    +      c    3    =      k    1    +      k    2    2  (      k    2    −      k    1    )  cos  ⁡  (  θ  )  sin  ⁡  (  θ  )  =      c    2      ⟹    (      k    2    −      k    1    )  sin  ⁡  2  θ  =      c    2    (      k    2    −      k    1    )  sin  ⁡  2  θ  +      k    1    +      k    2    =      c    1    +      c    2    +      c    3        k    2    (  sin  ⁡  2  θ  +  1  )  +      k    1    (  1  −  sin  ⁡  2  θ  )  =      c    1    +      c    2    +      c    3

Solving a set of 3 Nonlinear Equations In the following 3 equations: k 1 </msub>

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