Question Number 114409 by bemath last updated on 19/Sep/20

find solution set of equation   sin x = 2 , x ∈ C .

Answered by bobhans last updated on 19/Sep/20

sin x = ((e^(ix) −e^(−ix) )/(2i)) = 2   e^(ix) −e^(−ix)  = 4i , let e^(ix)  = z   z−z^(−1)  = 4i ; z^2 −4iz−1 = 0  z = ((4i±(√(−16+4)))/2) = (2±(√3))i   then e^(ix)  = (2±(√3))i = (2±(√3))e^(i.(π/2))   ix = ln (2±(√3)) +i((π/2)+2nπ); n∈Z  x=(4n+1)(π/2)±i ln (2+(√3)) ; n∈Z