A value of 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)))`

Options

A. `(pi)/(6)`
B. `sin^(-1)((Sqrt(3))/(4))`
C. `sin^(-1)((1)/(sqrt(3))`
D. `(pi)/(3)`

Answer

C

Explanation

`(2+ 3isin theta)/(1-2 sin theta) = (2+3i sin theta)/( 1-2i sin theta) xx(1+ 2i sin theta)/(1+ 2isin theta)`
`= ((2- 6 sin^(2) theta)+i( 7 sin theta))/(1 + 4 sin^(2) theta)`
For purely imaginary number
`sin^(2) theta = (1)/(3) or sin theta = (1)/(sqrt(3))`
`therefore theta= sin^(-1).(1)/(sqrt(3))`