Find the set of points belonging to the coordinate plane xy, for which the real part of the complex number ( 1 + i ) z 2 is positive.

Question

Find the set of points belonging to the coordinate plane xy, for which the real part of the complex number   (  1  +  i  )      z    2   is positive.

eurgylchnj · Accepted Answer

Step 1After having   cos  ⁡      (    2    θ    +          π      4        )    &amp;gt;  0 , you have  −            π      2        &amp;lt;      (    2    θ    +                  π        4              )    &amp;lt;            π      2      which is incorrect.To make it easy to understand why this is incorrect, let   α  :=  2  θ  +      π    4   .Then, we want to solve  cos  ⁡  α  &amp;gt;  0    and    −  π  ≤  θ  &amp;lt;  π  , ,i.e.  cos  ⁡  α  &amp;gt;  0    and    −      7    4    π  ≤  α  &amp;lt;      9    4    πwhich sould be easier to solve, to have  −      7    4    π  ≤  α  &amp;lt;  −      3    2    π    or    −      π    2    &amp;lt;  α  &amp;lt;      π    2      or        3    2    π  &amp;lt;  α  &amp;lt;      9    4    π  , ,i.e.  −  π  ≤  θ  &amp;lt;  −      7    8    π    or    −      3    8    π  &amp;lt;  θ  &amp;lt;      π    8      or        5    8    π  &amp;lt;  θ  &amp;lt;  πas always more elegant solutions are welcome(not sure if this is more elegant, but) another solution :Let   z  =  x  +  i  y where   x  ,  y  ∈      R   . Then,                    (1)                    ℜ        (        (        1        +        i        )                  z          2                )        =        ℜ        (        (        1        +        i        )        (        x        +        i        y                  )          2                )        =                  x          2                −                  y          2                −        2        x        y            When we solve       y    2    +  2  x  y  −      x    2    =  0 for y, we get  y  =  −  x  ±            x      2        +          x      2        =  −  x  ±      2       x  =  (  ±      2    −  1  )  xso from (1),                          ℜ        (        (        1        +        i        )                  z          2                )        &amp;gt;        0                                                  ⟺                (        (                  2                −        1        )        x        −        y        )        (        (                  2                +        1        )        x        +        y        )        &amp;gt;        0                                                  ⟺                −        (                  2                +        1        )        x        &amp;lt;        y        &amp;lt;        (                  2                −        1        )        x                or                (                  2                −        1        )        x        &amp;lt;        y        &amp;lt;        −        (                  2                +        1        )        x

Ximena Skinner · Accepted Answer

Step 1I would find the general solutions of the inequation first:                    cos        ⁡                              (                          2        θ        +                  π          4                                      )                          &amp;gt;        0                                    ⟺                −                  π          2                &amp;lt;        2        θ        +                  π          4                &amp;lt;                  π          2                        ⟺                −                              3            π                    4                &amp;lt;        2        θ        &amp;lt;                  π          4                                    mod                              2          π                                                          ⟺                −                              3            π                    8                &amp;lt;        θ        &amp;lt;                  π          8                                    mod                              π                    Now that if conventionally, we choose    −  π  &amp;lt;  θ  ≤  π , we obtain                                  ∙                            −                                            3              π                        8                                              &amp;lt;        θ                                  &amp;lt;                              π            8                          ,                                          ∙                            −        π                            &amp;lt;        θ                                  &amp;lt;        −                                            7              π                        8                          ,                                          ∙                                                                5              π                        8                                              &amp;lt;        θ                                  &amp;lt;        π        .

Find the set of points belonging to the coordinate plane

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