The complex number z = frac{i-1}{cos frac{pi} | vedclass

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Question

(A) Given $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$.Using Euler's formula,the denominator is $e^{i\pi/3}$.So,$z = (i-1) e^{-i\pi/3}$

Vedclass · Accepted Answer

$\sqrt{2} \left( \cos \frac{5 \pi}{12} + i \sin \frac{5 \pi}{12} \right)$

Vedclass · Answer

(A) Given $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$.Using Euler's formula,the denominator is $e^{i\pi/3}$.So,$z = (i-1) e^{-i\pi/3}$.We know $i-1 = \sqrt{2} \left( -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) = \sqrt{2} e^{i 3\pi/4}$.Therefore,$z = \sqrt{2} e^{i 3\pi/4} \cdot e^{-i\pi/3} = \sqrt{2} e^{i(3\pi/4 - \pi/3)} = \sqrt{2} e^{i(9\pi/12 - 4\pi/12)} = \sqrt{2} e^{i 5\pi/12}$.In polar form,this is $\sqrt{2} \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right)$.

The complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ is equal to $.....$

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