What is $(xy)^7=3$ in polar coordinates, in the form ____$=r^{14}$?

Question

This isn't a question of converting coordinates, and I've tried every version of $x=r\cos\theta$, $y=r\sin\theta$ I can think of.

So, what is $(xy)^7=3$ converted to an equation in polar coordinates. Of the form _____$=r^{14}$

Answer 1

$$3 = x^7 y^7 = r^{14}\cos^7(\theta)\sin^7(\theta) = $$ You have $$r^{14} = {3\over\cos^7(\theta)\sin^7(\theta)}.$$

Answer 2

You have $(xy)^7=(r\cos\theta\cdot r\sin\theta)^7=r^{14}\cos^7\theta \sin^7\theta $.

So your equation becomes $r^{14}\cos^7\theta \sin^7\theta=3$.

Which you can rearrange to the desired form.

Answer 3

Hint: Just substitute in $x=r \cos(\varphi)$ and $y=r \sin(\varphi)$, and express $r^{14}$.

Answer 4

$$(xy)^7=3$$

$$ (r\cos\theta. r\sin\theta )^7 =3 \rightarrow \dfrac{3}{(\cos ^7 \theta \sin ^7\theta )} = r^{14}$$

Or did you suppose it could be:

$$ \dfrac{3}{(\cos \theta ^7 \sin \theta ^7 )} = r^{14} ?$$