Question

A value of `theta` for which `(2+3isintheta)/(1-2isintheta)` purely imaginary, is : (1) `pi/3` (2) `pi/6` (3) `sin^(-1)((sqrt(3))/4)` (4) `sin^(-1)(1/(sqrt(3)))`

Answer 1

`(2+3isintheta)/(1-2isintheta) ** (1+2isintheta)/(1+2isintheta)`
`=(2+4isintheta+3isintheta+6i^2sin^2theta)/(1-4i^2sin^2theta)`
`=(2-6sin^2theta+7isintheta)/(1+4sin^2theta)`
`=(2-6sin^2theta) /(1+4sin^2theta) +i(7sintheta)/(1+4sin^2theta)`
For this number to be purely imaginary,
`(2-6sin^2theta) /(1+4sin^2theta) = 0`
`=> 2 - 6sin^2theta = 0`
`=>sin^2theta = 1/3`
`=> sintheta = 1/3`
`=> theta = sin^-1(1/sqrt3)`
So, option (4) is the correct option.