The sum of all possible values of theta in[-p | vedclass

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(B) Let $Z = \frac{1+i \cos 	heta}{1-2 i \cos 	heta}$.For $Z$ to be purely imaginary, the real part of $Z$ must be zero, or $Z + \overline{Z} = 0$.$

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$3$

Vedclass · Answer

(B) Let $Z = \frac{1+i \cos 	heta}{1-2 i \cos 	heta}$.For $Z$ to be purely imaginary, the real part of $Z$ must be zero, or $Z + \overline{Z} = 0$.$\frac{1+i \cos 	heta}{1-2 i \cos 	heta} + \frac{1-i \cos 	heta}{1+2 i \cos 	heta} = 0$$(1+i \cos 	heta)(1+2 i \cos 	heta) + (1-i \cos 	heta)(1-2 i \cos 	heta) = 0$$(1 + 3i \cos 	heta - 2 \cos^2 	heta) + (1 - 3i \cos 	heta - 2 \cos^2 	heta) = 0$$2 - 4 \cos^2 	heta = 0$$\cos^2 	heta = \frac{1}{2} \Rightarrow \cos 	heta = \pm \frac{1}{\sqrt{2}}$.For $	heta \in [-\pi, 2 \pi]$, the values of $	heta$ are $-\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$.The sum of these values is $(-\frac{3\pi}{4} - \frac{\pi}{4} + \frac{\pi}{4} + \frac{3\pi}{4} + \frac{5\pi}{4} + \frac{7\pi}{4}) = \frac{12\pi}{4} = 3\pi$.

The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary, is equal to (in $\pi$)

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