I've been trying to solve a problem that asks for the ratio of two area segments of a cardioid that's "cut" by a line. I think I've got a general idea of how to do it but I've run into a problem when trying to find the intersection points which I believe I need to get the bounds of integration.
One of the points was actually quite easy to find but the other is not. I'm stuck with the equation: $$(1-\cos(x))(\cos(x)+2\sin(x)) = 2$$ I know one of the points is at $\theta = \pi/2, r= 1$. I know it's possible to solve for it in cartesian but that requires some fairly advanced algebra and is a lengthy process as the root is not rational. So my question is: Is there a way to simplify the polar equation into something manageable that i'm simply failing to see?
Thank you.
Take $u = cos(x)$.
We have $(1-u)(u+2\sqrt{1-u^2}) = 2$.
Rearrange for $2\sqrt{1-u^2}(1-u)$ then square both sides.
We have $-4 u^4 + 8 u^3 - 8 u + 4 = u^4 - 2 u^3 + 5 u^2 - 4 u + 4$ You can see $u=0$ is a root here, which corresponds to $\cos(x) = 0$, $x = \frac{\pi}{2}$ for example.
This polynomial has another root $u \approx -0.40517$.