Number of solutions of this trigonometric equation.Q. Find the number of solutions of the equation sin ⁡ ( x ) + 2 sin ⁡ ( 2 x ) − sin ⁡ ( 3 x ) = 3, in the interval x ∈ ( 0 , π )I tried clubbing the sin ⁡ ( x ) and sin ⁡ ( 3 x ) terms together but got nothing. I also tried the sin ⁡ ( x ) with sin ⁡ ( 2 x ) and sin ⁡ ( 2 x ) with sin ⁡ ( 3 x ). How do i do it?

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Number of solutions of this trigonometric equation.Q. Find the number of solutions of the equation   sin  ⁡  (  x  )  +  2  sin  ⁡  (  2  x  )  −  sin  ⁡  (  3  x  )  =  3, in the interval   x  ∈  (  0  ,  π  )I tried clubbing the   sin  ⁡  (  x  ) and   sin  ⁡  (  3  x  ) terms together but got nothing. I also tried the   sin  ⁡  (  x  ) with   sin  ⁡  (  2  x  ) and   sin  ⁡  (  2  x  ) with   sin  ⁡  (  3  x  ). How do i do it?

rcampas4i · Accepted Answer

We have                    2        sin        ⁡        2        x        +        sin        ⁡        x        −        sin        ⁡        3        x                            =        4        sin        ⁡        x        cos        ⁡        x        −        2        sin        ⁡        x        cos        ⁡        2        x        =                                          =        sin        ⁡        x        (        4        cos        ⁡        x        −        4                  cos          2                ⁡        x        +        2        )        =                                          =        sin        ⁡        x                  [          3          −          (          2          cos          ⁡          x          −          1                      )            2                    ]                .            Since   sin  ⁡  x  ≥  0 in the interval   [  0  ,  π  ] and the second factor is bounded above by   3, we must have simultaneously   sin  ⁡  x  =  1,   2  cos  ⁡  x  =  1, which is not possible. Therefore the equation has no solutions in   [  0  ,  π  ]

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