Diagram of a multivaribale function

Question

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$

I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$. So I can conclude that, this inequation is valid, if $$\varphi \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]\cup\left[\frac{3\pi}{4},\frac{5\pi}{4}\right]$$

How I get this to paper?

Answer 1

Meanwhile I found a good way to draw this diagram.

You have to make a simple table with $\varphi \in [0, 2\pi]$ and the related radius of $\cos(2\varphi)$. Furthermore one have to look, where $\cos(2\varphi) \le 0$. There you get the interval I mentioned in the post above. Put these radii in the polar coordinatessystem but note, that the radius is positive where $\cos(2\varphi)$ is increasing and reversed.

Here is website programmed with GeoGebra to compare the results.