Question

Find the real values of θ for which the complex number $\frac{1+i\mathrm{cos\theta }}{1-2i\mathrm{cos\theta }}$ is purely real.

Answer 1

$$\begin{gathered} \frac{1+i\cos \theta }{1-2i\cos \theta } \\ =\frac{1+i\cos \theta }{1-2i\cos \theta }\times \frac{1+2i\cos \theta }{1+2i\cos \theta } \\ =\frac{1+2i\cos \theta +i\cos \theta -2\cos \theta }{1+4{\cos }^2\theta } \\ =\frac{1-2\cos \theta +i3\cos \theta }{1+4{\cos }^2\theta } \\ \mathrm{For}\mathrm{it}\mathrm{to}\mathrm{be}\mathrm{purely}\mathrm{real},\mathrm{the}\mathrm{imaginary}\mathrm{part}\mathrm{must}\mathrm{be}\mathrm{zero}. \\ 3\cos \theta =0 \\ \mathrm{This}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{odd}\mathrm{multiples}\mathrm{of}\frac{\mathrm{\pi }}{2}. \\ \therefore \mathrm{\theta }=\left(2n+1\right)\frac{\pi }{2},n\in Z \end{gathered}$$