How to solve s i n ( x ) = n for all n ∈ R ?I was wondering how one would solve the equation s i n ( x ) = n for all real n. You could of course use the Taylor series of s i n − 1 ( x ) but that wouldn't give an exact result.So I thought maybe you can use Euler's formula for s i n ( x ) ( s i n ( x ) = e i x − e − i x 2 i ) to get an exact result?

Question

How to solve   s  i  n  (  x  )  =  n for all   n  ∈            R      ?I was wondering how one would solve the equation   s  i  n  (  x  )  =  n for all real n. You could of course use the Taylor series of   s  i      n          −      1        (  x  ) but that wouldn&#039;t give an exact result.So I thought maybe you can use Euler&#039;s formula for   s  i  n  (  x  ) (  s  i  n  (  x  )  =                    e                  i          x                    −              e                  −          i          x                            2      i      ) to get an exact result?

x713x9o7r · Accepted Answer

To solve the equation       e          i      x        −      e          −      i      x        =  z you can define   w  =      e          i      x      , which gives  w  −      1    w    =  z    ⇔        w    2    −  z  w  −  1  =  0.Therefore,  w  =            z      ±                        z          2                +        4              2  and you can recover   x  =  −  i  log  ⁡  w. You need to be careful here since there are in fact may solutions, namely  x  =  −  i  log  ⁡  w  =  −  i  log  ⁡      |    w      |    +  i  (  arg  ⁡  w  +  2  k  π  )with   k  ∈      Z

How to solve sin(x)=n for all n in RR?

Answered question

Answer & Explanation

New Questions in Trigonometry