Can someone explain a polar coordinate conversion?

[sc5678]

I am having trouble understanding how (2x - x²)^1/2 becomes 2 cos θ.

Thanks

 

[HallsofIvy]

It doesn't. You can't just change a single expression into a particular coordinate system. In order to change to polar coordinates, you have to have an equation or function. Is this $y= (2x- x^2)^{1/2}$? If so, then you can start by squaring both sides: $y^2= 2x- x^2$ so that $x^2+ y^2= 2x$.

Now, $x^2+ y^2= r^2$ and $2x= 2r cos(\theta)$. The entire equation is now $r^2= 2r cos(\theta)$ and, dividing both sides by r, $r= 2 cos(\theta)$. But notice that this was reached by manipulating the whole equation- it was not just "$(2x- x^2)^{1/2}$" that became "$2 cos(\theta)$".