Find rectangular equation of a cardioid
Given the equation in polar form $$r = 1 - \sin\theta,$$ find the rectangular equation.
So far, I found:
$$x^2 + y^2 = 1 - 2\sin\theta + \sin^2\theta\quad x = \cos\theta - \sin\theta\cos\theta\quad y = \sin\theta - \sin^2\theta$$
Where should I go from here?
$\endgroup$ 11 Answer
$\begingroup$First, note that $y = r\sin{\theta} \Rightarrow \sin{\theta} = \frac{y}{r}$. This gives us: $$ r = 1 - \frac{y}{r} $$ Now, multiplying by $r$ on each side, we get: $$ r^2 = r - y $$ Noting that $r^2 = x^2 + y^2$, so this is the same as $$ x^2 + y^2 = \sqrt{x^2+y^2} - y $$ You can do what you wish from there to simplify it: I'm not sure what form you want it in.
$\endgroup$ 1
