How to Solve for x in the equation sin ⁡ ( 2 x ) = − 1 2 within the interval of [ 0 , 2 π ]?

Question

How to Solve for x in the equation   sin  ⁡  (  2  x  )  =  −      1    2   within the interval of   [  0  ,  2  π  ]?

Miguidi4y · Accepted Answer

If  sin  ⁡  (  2  x  )  =  −      1    2  then                    2        x                            =        arcsin        ⁡                  (          −                      1            2                    )                +        2        n        π                    2        x                            =        π        −        arcsin        ⁡                  (          −                      1            2                    )                +        2        m        π                            2        x                            =        −                  π          6                +        2        n        π                    2        x                            =        π        −                  (          −                      π            6                    )                +        2        m        π                            x                            =        −                  π          12                +        n        π                    2        x                            =                              7            π                    6                +        2        m        π                                        x                            =                              7            π                    12                +        m        π            where   m  ,  n  ∈      Z  . The requirement that   x  ∈  [  0  ,  2  π  ] implies n=1,2 and m=0,1

How to Solve for x in the equation sin ⁡ ( 2 x ) = &#x2212

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