So I need to find the POI (point of intersection) of the following two curves: \begin{align*} r & = 1 + \cos \theta, \\ r & = 2 - 2\cos \theta. \end{align*} What I did was I just set both the equations equal to each other... $$1 + \cos \theta = 2 - 2 \cos \theta$$ then isolated for $\theta$ and found the angle, which is $70^\circ$... however I drew the two graphs (looks like both cardioids) and it looks like I have 2 POI's... so I'm not sure why I only get one angle when I solve for $\theta$?


Hint: The $\cos$ function is positive in the first and fourth quadrant and negative in the second and third quadrant. So when you take the inverse of $\cos$ you need to take the fourth quadrant angle into account to get the other angle.

  • $\begingroup$ But how would I show that using what I did above? thats what im kinda confused about. $\endgroup$ – Nick Jun 10 '12 at 1:47
  • $\begingroup$ I don't have a calculator but the first angle would be $cos^{-1} (1/3)$ and the second would be $2\pi - cos^{-1}(1/3)$. $\endgroup$ – Eugene Jun 10 '12 at 1:48
  • $\begingroup$ Oh ok. Yeah i got the first angle as cos^-1 (1/3) .. so the other one i guess is just 2pi - cos^-1 (1/3) because the interval is 0 to 2pi? $\endgroup$ – Nick Jun 10 '12 at 1:58
  • $\begingroup$ @Nick Yup you got it. $\endgroup$ – Eugene Jun 10 '12 at 1:59

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