Question

Find the value of $\theta$ if $\frac{(3+2isin\theta )}{(1-2isin\theta )}$ is purely real or purely imaginary.

• A. $\theta =n\pi \pm \frac{\pi }{6},n\in Z$
• B. $\theta =n\pi \pm \frac{\pi }{2},n\in Z$
• C. $\theta =n\pi \pm \frac{\pi }{3},n\in Z$
• D. $\theta =n\pi \pm \frac{\pi }{4},n\in Z$

Answer 1

The correct option is C $\theta =n\pi \pm \frac{\pi }{3},n\in Z$

$z=\frac{3+2isin\theta }{1-2isin\theta }$
Multiplying numerator and denominator by conjugate, we get
$z=\frac{(3+2isin\theta )(1+2isin\theta )}{1+4si{n}^2\theta }=\frac{3-4si{n}^2\theta +8isin\theta }{1+4si{n}^2\theta }$
Now, $z$ is purely real if sin $\theta =0$ or $\theta =n\pi ,n\in Z$.
$z$ is puerly imaginary if $3-4si{n}^2\theta =0$
or $sin\theta =\pm \frac{\sqrt{3}}{2}=\pm sin\frac{\pi }{3}$
$⟹\theta =n\pi \pm \frac{\pi }{3},n\in Z$
Ans: C