The solutions of the equation sin ⁡ x + sin ⁡ 3 x = 8 3 3 are?

Question

The solutions of the equation   sin  ⁡      x    +  sin  ⁡      3    x    =      8          3              3             are?

Clarence Terrell · Accepted Answer

HINTTo begin with, notice that   sin  ⁡  (  3  x  )  =  3  sin  ⁡  (  x  )  −  4      sin          3        ⁡  (  x  ). Hence it results that:                    sin        ⁡        (        x        )        +        sin        ⁡        (        3        x        )        =                  8                      3                          3                                                          ⟺        4        sin        ⁡        (        x        )        −        4                  sin                      3                          ⁡        (        x        )        =                  8                      3                          3                                                                                      ⟺        sin        ⁡        (        x        )        −                  sin                      3                          ⁡        (        x        )        =                              2                          3                                9                    By inspection, one concludes that                    sin        ⁡        (        x        )        =                  1                      3                              satisfies the resulting equation.From then on, you can factor the cubic equation to obtain a quadratic which is easy to deal with.

ucatzABI · Accepted Answer

As you obtained,       t    −          t      3        =          2              3                  3                           ,    where     t    =    sin    ⁡    x        t    −          t      3        =          1              3              −          1              3                  3                          ⟹                      (                  1                      3                          )            3        −          t      3        =          (              1                  3                    −      t      )      Now using the fact that       a    3    −      b    3    =  (  a  −  b  )  (      a    2    +      b    2    +  a  b  )One of the obvious solution is       t  =            1              3            If       t  ≠            1              3              ,, we have                 t      2        +          1      3        +          t              3              =    1  i.e.                         (        t        +                  1                      2                          3                                      )            2        =          3      4            . That gives       t    =          1              3              ,    −          2              3            As       −            2              3              &amp;lt;  −  1, that is not a valid solution. So the only solution is   sin  ⁡  x  =            1              3

The solutions of the equation sin x+sin 3x=(8)/(3 sqrt(3)) are?

Answered question

Answer & Explanation

New Questions in Trigonometry