The sum of all possible values of [- ,2 ], for which (1+i )/(1-2i ) is purely imaginary, i…

Question

The sum of all possible values of $	heta \in [-\pi ,2\pi ]$, for which $\frac{1+i\cos 	heta }{1-2i\cos 	heta }$ is purely imaginary, is equal to :

Accepted Answer

$\frac{1+i\cos 	heta }{1-2i\cos 	heta }$ is purely imaginary

$n=\frac{1+i\cos 	heta }{1-2i\cos 	heta }	imes \frac{1+2i\cos 	heta }{1+2i\cos 	heta }=\frac{1+3i\cos 	heta -2{\cos }^{2}	heta }{1+4{\cos }^{2}	heta }$

$n=\frac{1-2{\cos }^{2}	heta }{1+4{\cos }^{2}	heta }+i(\frac{3\cos 	heta }{1+4{\cos }^{2}	heta })$

$n$ is purely imaginary

$\Rightarrow \frac{1-2{\cos }^{2}	heta }{1+4{\cos }^{2}	heta }=0$

$\Rightarrow {\cos }^{2}	heta =\frac{1}{2}$

$\Rightarrow \cos 	heta =\pm \frac{1}{\sqrt{2}}$

$	heta $ can be $\frac{\pi }{4},\frac{-\pi }{4},\frac{3\pi }{4},\frac{-3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$

Sum of all possible values of $	heta =3\pi $

Solutions · 1