Let A = θ ∈ - π 2 , π : 3 + 2 i   s i n θ 1 - 2 i s i n θ i s p u r e l y i m a g i n a r y . Then the sum of the elements in A is:

Step-by-step solution

Given complex number is z = 3 + 2 i s i n θ 1 - 2 i s i n θ ⇒ z = ( 3 + 2 i s i n θ ) ( 1 + 2 i s i n θ ) ( 1 + 4 s i n 2 θ ) Since given complex number is purely imaginary ⇒ R e ( z ) = 0 ⇒ 3 - 4 s i n 2 θ = 0 ⇒ s i n θ = ± 3 2 ⇒ θ = - π 3 , π 3 , 2 π 3 ⇒ sum of all possible values of θ = 2 π 3 .