91Ó°ÊÓ

Any complex number z can be represented in the polar form: \[z = re^{i\theta}\] where r is the magnitude of z and \(\theta\) is the argument of z. Now, we need to find the real part of \(z^3\).

Using the polar form of z, we can compute \(z^3\) as follows: \[z^3 = (re^{i\theta})^3 = r^3e^{3i\theta}\]

To find the real part of \(z^3\), we can rewrite it in its rectangular form using Euler's formula: \[e^{3i\theta} = \cos(3\theta) + i\sin(3\theta)\] So, the rectangular form of \(z^3\) is: \[z^3 = r^3(\cos(3\theta) + i\sin(3\theta))\] The real part of \(z^3\) is: \[Re(z^3) = r^3\cos(3\theta)\]

We want to find the subset of complex numbers for which the real part of \(z^3\) is positive, i.e.: \[r^3\cos(3\theta) > 0\] Since \(r^3\) is always positive, we only need to consider when \(\cos(3\theta) > 0\). The cosine function is positive in the first and fourth quadrants, which corresponds to angles \(\theta\) in the ranges: \[\frac{-\pi}{3} < \theta < \frac{\pi}{3} \text{ and }\frac{5\pi}{3} < \theta < \frac{7\pi}{3}\] (imagine three \(\theta\) intervals for \(3\theta\) instead of one) Thus, the subset of the complex plane consisting of complex numbers z such that the real part of \(z^3\) is a positive number is the set of all complex numbers for which the argument \(\theta\) lies in the following intervals: \[\frac{-\pi}{3} < \theta < \frac{\pi}{3} \text{ and }\frac{5\pi}{3} < \theta < \frac{7\pi}{3}\]