Dear Uncle Colin,

I’m told that $x \sin^3(\theta) + y \cos^3(\theta) =\sin(\theta)\cos(\theta)$ and that $x\sin(\theta) = y\cos(\theta)$. I need to work out $x^2+y^2$. How would you approach it?

- Simply Impossible, Need Explanation

Hi, SINE, and thanks for your message! I certainly hope it isn’t simply impossible.

I’d probably start with some replacements. Let’s start by replacing the $x$ and once of the sines in the first equation with $y\cos(\theta)$:

  • $y \cos(\theta)\sin^2(\theta) + y\cos^3(\theta) = \sin(\theta)\cos(\theta)$. Now group:
  • $y \cos(\theta)\left( \sin^2(\theta)+ \cos^2(\theta)\right) = \sin(\theta)\cos(\theta)$ – but that factor is 1!
  • $y \cos(\theta) = \sin(\theta)\cos(\theta)$, which means either $\cos(\theta)=0$ or $y = \sin(\theta)$.

Case 1: if $\cos(\theta)=0$, then $x=0$ and $y$ could be anything at all.

Case 2, the more interesting one, gives $x = \cos(\theta)$. That gives $x^2 + y^2 = \cos^2(\theta) + \sin^2(\theta)=1$.

Hope that helps!

- Uncle Colin