Write function from polar to rectangular coordinates.

Question

I need to write this functions in rectangular coordinates: $$f(r,\theta)=r^{2k+5}\cos5\theta$$ $$g(r,\theta)=r^{2k+5}\cos5\theta$$ Of course the radius is very easy to convert to $x$ and $y$. The problem after is the sine and cosine because that $5$ inside is very disturbing. I don't know if there is any kind of trick to do this.

Thanks.

Answer 1

Write $x+iy=re^{i\theta}$, then $(x+iy)^5=r^5e^{5i\theta}$, so $$r^5\cos5\theta={\rm Re}(r^5e^{5i\theta})={\rm Re}((x+iy)^5)=x^5+10x^3(iy)^2+5x(iy)^4=x^5-10x^3y^2+5xy^4,$$ and also $$r^5\sin5\theta={\rm Im}(r^5e^{5i\theta})={\rm Im}((x+iy)^5)=5x^4y-10x^2y^3+y^5.$$ Now, use also that $r^2=x^2+y^2$.