Find real theta such that frac{3+2 i sin thet | vedclass

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Question

(A) Let $z = \frac{3+2 i \sin 	heta}{1-2 i \sin 	heta}$.Multiplying the numerator and denominator by the conjugate of the denominator $(1+2 i \sin \

Vedclass · Accepted Answer

$	heta = n\pi, n \in Z$

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(A) Let $z = \frac{3+2 i \sin 	heta}{1-2 i \sin 	heta}$.Multiplying the numerator and denominator by the conjugate of the denominator $(1+2 i \sin 	heta)$:$z = \frac{(3+2 i \sin 	heta)(1+2 i \sin 	heta)}{(1-2 i \sin 	heta)(1+2 i \sin 	heta)}$$z = \frac{3 + 6 i \sin 	heta + 2 i \sin 	heta + 4 i^2 \sin^2 	heta}{1^2 - (2 i \sin 	heta)^2}$Since $i^2 = -1$,we have:$z = \frac{3 + 8 i \sin 	heta - 4 \sin^2 	heta}{1 + 4 \sin^2 	heta}$$z = \frac{3 - 4 \sin^2 	heta}{1 + 4 \sin^2 	heta} + i \frac{8 \sin 	heta}{1 + 4 \sin^2 	heta}$For $z$ to be purely real,the imaginary part must be zero:$\frac{8 \sin 	heta}{1 + 4 \sin^2 	heta} = 0$This implies $\sin 	heta = 0$.Therefore,$	heta = n\pi$ for any integer $n \in Z$.

Find real $\theta$ such that $\frac{3+2 i \sin \theta}{1-2 i \sin \theta}$ is purely real.

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