Question

for what value of thita is the expression 5+2icos thita/4-3icos thita purely real?

Answer 1

$$\begin{gathered} \frac{5+2\mathrm{i}\cos \mathrm{\theta }}{4-3\mathrm{i}\cos \mathrm{\theta }} \\ =\frac{5+2\mathrm{i}\cos \mathrm{\theta }}{4-3\mathrm{i}\cos \mathrm{\theta }}\times \frac{4+3\mathrm{i}\cos \mathrm{\theta }}{4+3\mathrm{i}\cos \mathrm{\theta }} \\ =\frac{20+15\mathrm{i}\cos \mathrm{\theta }+8\mathrm{i}\cos \mathrm{\theta }-6{\cos }^2\mathrm{\theta }}{16+9{\cos }^2\mathrm{\theta }} \\ =\frac{20-6{\cos }^2\mathrm{\theta }}{16+9{\cos }^2\mathrm{\theta }}+\frac{23\mathrm{i}\cos \mathrm{\theta }}{16+9{\cos }^2\mathrm{\theta }} \\ \mathrm{Since}\mathrm{complex}\mathrm{number}\mathrm{is}\mathrm{purely}\mathrm{real},\mathrm{then} \\ \frac{23\mathrm{i}\cos \mathrm{\theta }}{16+9{\cos }^2\mathrm{\theta }} \\ \Rightarrow \cos \mathrm{\theta }=0 \\ \Rightarrow \mathrm{\theta }=\left(2\mathrm{n}+1\right)\frac{\mathrm{\pi }}{2};\mathrm{n}\in \mathrm{Z} \end{gathered}$$