The complex number z = (i - 1)/(cos(π)/(3) + isin(π)/(3)) is equal...

Given,

$z = \dfrac{i - 1}{cos\dfrac{\pi}{3} + isin\dfrac{\pi}{3}}$

$⇒ z = \dfrac{\sqrt{2}\left(\dfrac{- 1}{\sqrt{2}} + \dfrac{i}{\sqrt{2}}\right)}{cos\dfrac{\pi}{3} + isin\dfrac{\pi}{3}}$

Now argument of complex number $\dfrac{- 1}{\sqrt{2}} + \dfrac{i}{\sqrt{2}}$ is $\dfrac{3\pi}{4}$

So, by euler's formula ${e}^{i\theta} = cos\theta + isin\theta$ we get,

$⇒ z = \dfrac{\sqrt{2}{e}^{i3\pi / 4}}{{e}^{i\pi / 3}}$

$⇒ z = \sqrt{2}{e}^{i5\pi / 12}$

$⇒ z = \sqrt{2}\left(cos\dfrac{5\pi}{12} + isin\dfrac{5\pi}{12}\right)$