If real numbers x and y fit the equality 2 x + i y + i = 1 + i sin ⁡ α 1 + i sin ⁡ 3 α , what does x y equal?

Question

If real numbers x and y fit the equality             2      x      +      i              y      +      i        =            1      +      i      sin      ⁡      α              1      +      i      sin      ⁡      3      α      , what does       x    y   equal?

dariajoq9 · Accepted Answer

Here is a slightly simpler solution.  (  2  x  +  i  )  (  1  +  i  sin  ⁡      3    α    )  =  (  y  +  i  )  (  1  +  i  sin  ⁡      α    )Now let us equate the imaginary terms.  i  (  1  +  2  x  sin  ⁡      3    α    )  =  i  (  1  +  y  sin  ⁡      α    )  i  (  1  +  2  x  sin  ⁡      3    α    )  =  i  (  1  +  y  sin  ⁡      α    )      y    x    =            2      sin      ⁡              3        α                    sin      ⁡              α                  y    x    =  2  (  2      cos    2    ⁡      α    +      cos    2    ⁡      α    −      sin    2    ⁡      α    )      y    x    =  2  (  1  +  2      cos    2    ⁡      α    −  2      sin    2    ⁡      α    )We know that   cos  ⁡      2    x    =      cos    2    ⁡      x    −      sin    2    ⁡      x  , so we get.      y    x    =  2  +  4  cos  ⁡      2    α

Jaycee Mathis · Accepted Answer

2        x        +        i                    y        +        i              =                    −                  i          2                +        i        sin        ⁡                  α                            −                  i          2                +        i        sin        ⁡                  3          α                      =                    sin        ⁡                  α                −        i                    sin        ⁡                  3          α                −        i              =                    2        x        +        sin        ⁡        α                    y        +        sin        ⁡                  3          α                    where we added numerators and denominators separatelyCross multiply and equate real/imaginary parts  y  +  sin  ⁡      3    α    =  2  x  +  sin  ⁡      α    2  x  sin  ⁡      3    α    =  y  sin  ⁡      α  First equation is not useful as it gives a linear relation.But quotient relation in second equation            x      y        =                    sin        ⁡                  α                            2        sin        ⁡                  3          α                    gives the needed result.

If real numbers x and y fit the equality 2 x + i </mrow>

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