How to sketch the following set?

Question

Sketch on the argand's diagram the following set: $A=\{w \in \mathbb{C} \backslash \{0\}:w^3-w^{-3} \in \mathbb{R}\}$. How to approach this question?

Answer 1

One approach is as follows:

Defining $w = x + iy$, find an expression for $\text{Im}[w^3 - w^{-3}]$ in terms of $x$ and $y$. Set the result equal to $0$.

In particular, we have $$ w^3 - w^{-3} = (x + iy)^3 - \frac{(x - iy)^3}{(x^2 + y^2)^3} $$ Alternatively, defining $w = r \cos \theta + i r \sin \theta$, we have $$ w^3 - w^{-3} = r^3 \cos 3 \theta + i r^3 \sin 3 \theta - \frac{\cos 3\theta - i \sin 3\theta}{r^3} $$

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Setting the imaginary part of the above equal to zero gives us $$ r^3 \sin 3 \theta - r^{-3}\sin 3 \theta = 0 \implies\\ (r^6 - 1) \sin 3 \theta = 0 $$ So, the solution will be the circle $|w| = 1$, along with the lines associated with the arguments $ \pi k/ 3$ where $k \in \{0,1,\dots,6\}$.