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Question: If \(300^{o}\), then the difference in the amplitudes of \(arg(z) = \theta\) and \(arg(\overline{z})...

If 300o300^{o}, then the difference in the amplitudes of arg(z)=θarg(z) = \theta and arg(z‾)=arg(\overline{z}) = is.

A

θ\theta

B

−θ- \theta

C

π−θ\pi - \theta

D

0

Answer

π−θ\pi - \theta

Explanation

Solution

Squaring the given relations implies that

θ=π10\theta = \frac{\pi}{10}

Now z=−1−i3z = - 1 - i\sqrt{3}

α=tan⁡−1∣ba∣=tan⁡−1∣−31∣=π3θ=−(π−α)=−(π−π/3)=−2π3\alpha = \tan^{- 1}\left| \frac{b}{a} \right| = \tan^{- 1}\left| - \frac{\sqrt{3}}{1} \right| = \frac{\pi}{3}\theta = - (\pi - \alpha) = - (\pi - \pi/3) = \frac{- 2\pi}{3}.