If x_r = cos(pi/3^r) - isin(pi/3^r) (where i | vedclass

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(C) Given $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$.We need to find the product $P = x_1 \cdot x_2 \cdot x_3 \cdots \infty$.Using the property of complex

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$-i$

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(C) Given $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$.We need to find the product $P = x_1 \cdot x_2 \cdot x_3 \cdots \infty$.Using the property of complex numbers,$e^{-i	heta} = \cos 	heta - i\sin 	heta$,we have $x_r = e^{-i\pi/3^r}$.Thus,$P = e^{-i\pi/3} \cdot e^{-i\pi/9} \cdot e^{-i\pi/27} \cdots \infty$.$P = e^{-i\pi (1/3 + 1/9 + 1/27 + \cdots \infty)}$.The exponent is a geometric series with first term $a = 1/3$ and common ratio $r = 1/3$.The sum of the infinite geometric series is $S = a / (1 - r) = (1/3) / (1 - 1/3) = (1/3) / (2/3) = 1/2$.Therefore,$P = e^{-i\pi(1/2)} = e^{-i\pi/2}$.Using Euler's formula,$e^{-i\pi/2} = \cos(\pi/2) - i\sin(\pi/2) = 0 - i(1) = -i$.

If $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$ (where $i = \sqrt{-1}$),then the value of $x_1 \cdot x_2 \cdot x_3 \cdots \infty$ is:

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