First slide

Algebra of complex numbers

Question

The real value of $' θ'$ for which the expression $\frac{3 + 2 i \sin θ}{1 - 2 i \sin θ}$ is purely imaginary is

Moderate

A

$n π, n \in Z$
B

$n π \pm \frac{π}{2}, n \in Z$
C

$n π \pm \frac{π}{3}, n \in Z$
D

$n π \pm \frac{π}{6}, n \in Z$

Solution

$\frac{3 + 2 i \sin θ}{1 - 2 i \sin θ}$ is purely imaginary
$\Rightarrow$ Real part is zero

$∴ \frac{3 - 4 \sin^{2} θ}{1 + {(2 \sin θ)}^{2}} = 0$

$\Rightarrow \sin^{2} θ = \frac{3}{4}$

$\Rightarrow \sin^{2} θ = \pm \frac{\sqrt{3}}{2}$

$∴ = \pm \sin \frac{π}{3}$

$∴$ Solution is $θ = n π \pm \frac{π}{3}, n \in z$

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