I'm trying to do this problem but I cannot get it. Here is what I have done so far: Start with $$r=1 – \cos \theta \;\;\;\text{with} \;\;\; 0 \le \theta \le 2 \pi$$ So I multiply both sides by r: $${r^2} = r-r\cos \theta$$ Now I have: $${x^2+y^2}=r-x$$ Correct?
Next: $${x^2+y^2}=\sqrt{x^2+y^2}-x$$ And that is where I'm stuck. What should I do? Does anyone know the final answer so I know what I am heading for?
Thanks.
Best Answer
As indicated in the comments, it doesn’t seem this can be reduced any further. It would help to notice that this curve is a limaçon, which is partially why I know you can’t do much more with it.
The curve can be parametrized, however, as $$\left\{ \begin{array}{} x = (1-\cos t)\cos t \\ y = (1-\sin t)\sin t \\ \end{array} \right.$$
To read more about equations for limaçons, turn here.
Summary of the comments: As a student, you have demonstrated that you understand the relationship between polar and rectangular coordinates, and trying to write this expression as a piecewise function $x\mapsto y$ is absurd.