Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$.
(a) Find the intersection points of the curve with the line $\theta = \pi/4$.
(b) Find the intersection points of the curve with the circle $r = 1/2$.
(c) Find the slope of the tangent line to the curve at the point $(3/2,−2\pi/3)$.
(d) Find the length of the part of the curve in the fourth quadrant.
Best Answer
For (a), it's worth noting that $\theta=\pi/4$, $\theta=-3\pi/4$, and $r=0$ each put a point $(r,\theta)$ on the "line" $\theta=\pi/4$.
For (b), you need only find $-\pi\leq\theta\leq\pi$ such that $\frac12=1-\cos\theta$.
For (c), do you know how to find the tangent line at a point on a polar curve?
For (d), do you know how to find arc lengths?