Question

Change to Cartesian coordinates the equation : sqr root(r)=sqr root(a)*cos(θ/2)

Answer 1

\[\sqrt{r}=\sqrt{acos(\theta/2)}\]So,\[r=acos(\theta/2)\]In polar coordinates\[x=rcos \theta\]Thus,\[x/\cos \theta=acos(\theta/2)\]\[x=acos (\theta/2)\cos \theta\]You can re-write this using double-angle identities, etc if you want. Solving for y is similar.