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 [itex]y= (2x- x^2)^{1/2}[/itex]? If so, then you can start by squaring both sides: [itex]y^2= 2x- x^2[/itex] so that [itex]x^2+ y^2= 2x[/itex].

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